diff --git a/benchmarks/SimpleHandwrittenPDE/Manifest.toml b/benchmarks/SimpleHandwrittenPDE/Manifest.toml index 211b61baa..45faffceb 100644 --- a/benchmarks/SimpleHandwrittenPDE/Manifest.toml +++ b/benchmarks/SimpleHandwrittenPDE/Manifest.toml @@ -1,19 +1,19 @@ # This file is machine-generated - editing it directly is not advised -julia_version = "1.9.1" +julia_version = "1.10.10" manifest_format = "2.0" -project_hash = "f525cc990457ba8862f03bbdc865a92251324e1e" +project_hash = "ed974a7a662e51b1521b02ca6a2b78592e347892" [[deps.ADTypes]] -git-tree-sha1 = "a4c8e0f8c09d4aa708289c1a5fc23e2d1970017a" +git-tree-sha1 = "7927b9af540ee964cc5d1b73293f1eb0b761a3a1" uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" -version = "0.2.1" +version = "1.16.0" +weakdeps = ["ChainRulesCore", "ConstructionBase", "EnzymeCore"] -[[deps.AbstractAlgebra]] -deps = ["GroupsCore", "InteractiveUtils", "LinearAlgebra", "MacroTools", "Preferences", "Random", "RandomExtensions", "SparseArrays", "Test"] -git-tree-sha1 = "41b45260ad00b666558c16f7cd9db0e8af408ea1" -uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d" -version = "0.31.1" + [deps.ADTypes.extensions] + ADTypesChainRulesCoreExt = "ChainRulesCore" + ADTypesConstructionBaseExt = "ConstructionBase" + ADTypesEnzymeCoreExt = "EnzymeCore" [[deps.AbstractFFTs]] deps = ["LinearAlgebra"] @@ -26,84 +26,72 @@ weakdeps = ["ChainRulesCore", "Test"] AbstractFFTsChainRulesCoreExt = "ChainRulesCore" AbstractFFTsTestExt = "Test" -[[deps.AbstractTrees]] -git-tree-sha1 = "faa260e4cb5aba097a73fab382dd4b5819d8ec8c" -uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c" -version = "0.4.4" +[[deps.Accessors]] +deps = ["CompositionsBase", "ConstructionBase", "Dates", "InverseFunctions", "MacroTools"] +git-tree-sha1 = "3b86719127f50670efe356bc11073d84b4ed7a5d" +uuid = "7d9f7c33-5ae7-4f3b-8dc6-eff91059b697" +version = "0.1.42" + + [deps.Accessors.extensions] + AxisKeysExt = "AxisKeys" + IntervalSetsExt = "IntervalSets" + LinearAlgebraExt = "LinearAlgebra" + StaticArraysExt = "StaticArrays" + StructArraysExt = "StructArrays" + TestExt = "Test" + UnitfulExt = "Unitful" + + [deps.Accessors.weakdeps] + AxisKeys = "94b1ba4f-4ee9-5380-92f1-94cde586c3c5" + IntervalSets = "8197267c-284f-5f27-9208-e0e47529a953" + LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" + StaticArrays = "90137ffa-7385-5640-81b9-e52037218182" + StructArrays = "09ab397b-f2b6-538f-b94a-2f83cf4a842a" + Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" + Unitful = "1986cc42-f94f-5a68-af5c-568840ba703d" [[deps.Adapt]] deps = ["LinearAlgebra", "Requires"] -git-tree-sha1 = "76289dc51920fdc6e0013c872ba9551d54961c24" +git-tree-sha1 = "f7817e2e585aa6d924fd714df1e2a84be7896c60" uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" -version = "3.6.2" -weakdeps = ["StaticArrays"] +version = "4.3.0" +weakdeps = ["SparseArrays", "StaticArrays"] [deps.Adapt.extensions] + AdaptSparseArraysExt = "SparseArrays" AdaptStaticArraysExt = "StaticArrays" -[[deps.ApproxFun]] -deps = ["AbstractFFTs", "ApproxFunBase", "ApproxFunFourier", "ApproxFunOrthogonalPolynomials", "ApproxFunSingularities", "Calculus", "DomainSets", "FastTransforms", "LinearAlgebra", "RecipesBase", "Reexport", "SpecialFunctions", "StaticArrays"] -git-tree-sha1 = "eba77cc445bf6e39e3ae63904a1001a97aad66e8" -uuid = "28f2ccd6-bb30-5033-b560-165f7b14dc2f" -version = "0.13.23" -weakdeps = ["DualNumbers"] - - [deps.ApproxFun.extensions] - ApproxFunDualNumbersExt = "DualNumbers" - -[[deps.ApproxFunBase]] -deps = ["AbstractFFTs", "BandedMatrices", "BlockArrays", "BlockBandedMatrices", "Calculus", "Combinatorics", "DSP", "DomainSets", "DualNumbers", "FFTW", "FillArrays", "InfiniteArrays", "IntervalSets", "LazyArrays", "LinearAlgebra", "LowRankMatrices", "SparseArrays", "SpecialFunctions", "StaticArrays", "Statistics"] -git-tree-sha1 = "0040e8cbee9492cfc74d3da3ea6fb92f05404db3" -uuid = "fbd15aa5-315a-5a7d-a8a4-24992e37be05" -version = "0.9.12" - -[[deps.ApproxFunFourier]] -deps = ["AbstractFFTs", "ApproxFunBase", "BandedMatrices", "DomainSets", "FFTW", "FastTransforms", "InfiniteArrays", "IntervalSets", "LinearAlgebra", "Reexport", "StaticArrays"] -git-tree-sha1 = "872f236d5a0c6dd69c41517ab2471a6755aeacf4" -uuid = "59844689-9c9d-51bf-9583-5b794ec66d30" -version = "0.3.23" - -[[deps.ApproxFunOrthogonalPolynomials]] -deps = ["ApproxFunBase", "BandedMatrices", "BlockArrays", "BlockBandedMatrices", "DomainSets", "FastGaussQuadrature", "FastTransforms", "FillArrays", "HalfIntegers", "IntervalSets", "LinearAlgebra", "OddEvenIntegers", "Reexport", "SpecialFunctions", "StaticArrays"] -git-tree-sha1 = "71219dd7a6ea7086f081eb2f99b238f70cb50189" -uuid = "b70543e2-c0d9-56b8-a290-0d4d6d4de211" -version = "0.6.45" -weakdeps = ["Static"] - - [deps.ApproxFunOrthogonalPolynomials.extensions] - ApproxFunOrthogonalPolynomialsStaticExt = "Static" - -[[deps.ApproxFunSingularities]] -deps = ["ApproxFunBase", "ApproxFunOrthogonalPolynomials", "DomainSets", "HalfIntegers", "IntervalSets", "LinearAlgebra", "OddEvenIntegers", "Reexport", "SpecialFunctions"] -git-tree-sha1 = "3b0033b1d67dd7a9c05d51932e5c6b30946fb89a" -uuid = "f8fcb915-6b99-5be2-b79a-d6dbef8e6e7e" -version = "0.3.16" -weakdeps = ["StaticArrays"] +[[deps.AliasTables]] +deps = ["PtrArrays", "Random"] +git-tree-sha1 = "9876e1e164b144ca45e9e3198d0b689cadfed9ff" +uuid = "66dad0bd-aa9a-41b7-9441-69ab47430ed8" +version = "1.1.3" - [deps.ApproxFunSingularities.extensions] - ApproxFunSingularitiesStaticArraysExt = "StaticArrays" +[[deps.ArgCheck]] +git-tree-sha1 = "f9e9a66c9b7be1ad7372bbd9b062d9230c30c5ce" +uuid = "dce04be8-c92d-5529-be00-80e4d2c0e197" +version = "2.5.0" [[deps.ArgTools]] uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" version = "1.1.1" -[[deps.ArnoldiMethod]] -deps = ["LinearAlgebra", "Random", "StaticArrays"] -git-tree-sha1 = "62e51b39331de8911e4a7ff6f5aaf38a5f4cc0ae" -uuid = "ec485272-7323-5ecc-a04f-4719b315124d" -version = "0.2.0" - [[deps.ArrayInterface]] -deps = ["Adapt", "LinearAlgebra", "Requires", "SparseArrays", "SuiteSparse"] -git-tree-sha1 = "f83ec24f76d4c8f525099b2ac475fc098138ec31" +deps = ["Adapt", "LinearAlgebra"] +git-tree-sha1 = "9606d7832795cbef89e06a550475be300364a8aa" uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9" -version = "7.4.11" +version = "7.19.0" [deps.ArrayInterface.extensions] ArrayInterfaceBandedMatricesExt = "BandedMatrices" ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices" ArrayInterfaceCUDAExt = "CUDA" + ArrayInterfaceCUDSSExt = "CUDSS" + ArrayInterfaceChainRulesCoreExt = "ChainRulesCore" + ArrayInterfaceChainRulesExt = "ChainRules" ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore" + ArrayInterfaceReverseDiffExt = "ReverseDiff" + ArrayInterfaceSparseArraysExt = "SparseArrays" ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore" ArrayInterfaceTrackerExt = "Tracker" @@ -111,21 +99,20 @@ version = "7.4.11" BandedMatrices = "aae01518-5342-5314-be14-df237901396f" BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0" CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba" + CUDSS = "45b445bb-4962-46a0-9369-b4df9d0f772e" + ChainRules = "082447d4-558c-5d27-93f4-14fc19e9eca2" + ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527" + ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" + SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c" Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c" -[[deps.ArrayInterfaceCore]] -deps = ["LinearAlgebra", "SnoopPrecompile", "SparseArrays", "SuiteSparse"] -git-tree-sha1 = "e5f08b5689b1aad068e01751889f2f615c7db36d" -uuid = "30b0a656-2188-435a-8636-2ec0e6a096e2" -version = "0.1.29" - [[deps.ArrayLayouts]] -deps = ["FillArrays", "LinearAlgebra"] -git-tree-sha1 = "dcda7e0ac618210eabf43751d5cafde100dd539b" +deps = ["FillArrays", "LinearAlgebra", "StaticArrays"] +git-tree-sha1 = "120e392af69350960b1d3b89d41dcc1d66543858" uuid = "4c555306-a7a7-4459-81d9-ec55ddd5c99a" -version = "1.3.0" +version = "1.11.2" weakdeps = ["SparseArrays"] [deps.ArrayLayouts.extensions] @@ -134,127 +121,156 @@ weakdeps = ["SparseArrays"] [[deps.Artifacts]] uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" -[[deps.AxisAlgorithms]] -deps = ["LinearAlgebra", "Random", "SparseArrays", "WoodburyMatrices"] -git-tree-sha1 = "66771c8d21c8ff5e3a93379480a2307ac36863f7" -uuid = "13072b0f-2c55-5437-9ae7-d433b7a33950" -version = "1.0.1" +[[deps.AutoHashEquals]] +git-tree-sha1 = "4ec6b48702dacc5994a835c1189831755e4e76ef" +uuid = "15f4f7f2-30c1-5605-9d31-71845cf9641f" +version = "2.2.0" [[deps.BandedMatrices]] deps = ["ArrayLayouts", "FillArrays", "LinearAlgebra", "PrecompileTools"] -git-tree-sha1 = "0b816941273b5b162be122a6c94d706e3b3125ca" +git-tree-sha1 = "e35c672b239c5105f597963c33e740eeb46cf0ab" uuid = "aae01518-5342-5314-be14-df237901396f" -version = "0.17.38" -weakdeps = ["SparseArrays"] +version = "1.9.4" [deps.BandedMatrices.extensions] BandedMatricesSparseArraysExt = "SparseArrays" + CliqueTreesExt = "CliqueTrees" + + [deps.BandedMatrices.weakdeps] + CliqueTrees = "60701a23-6482-424a-84db-faee86b9b1f8" + SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" [[deps.Base64]] uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" -[[deps.Bijections]] -git-tree-sha1 = "fe4f8c5ee7f76f2198d5c2a06d3961c249cce7bd" -uuid = "e2ed5e7c-b2de-5872-ae92-c73ca462fb04" -version = "0.1.4" +[[deps.Bessels]] +git-tree-sha1 = "4435559dc39793d53a9e3d278e185e920b4619ef" +uuid = "0e736298-9ec6-45e8-9647-e4fc86a2fe38" +version = "0.2.8" [[deps.BitFlags]] -git-tree-sha1 = "43b1a4a8f797c1cddadf60499a8a077d4af2cd2d" +git-tree-sha1 = "0691e34b3bb8be9307330f88d1a3c3f25466c24d" uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35" -version = "0.1.7" +version = "0.1.9" [[deps.BitTwiddlingConvenienceFunctions]] deps = ["Static"] -git-tree-sha1 = "0c5f81f47bbbcf4aea7b2959135713459170798b" +git-tree-sha1 = "f21cfd4950cb9f0587d5067e69405ad2acd27b87" uuid = "62783981-4cbd-42fc-bca8-16325de8dc4b" -version = "0.1.5" +version = "0.1.6" [[deps.BlockArrays]] deps = ["ArrayLayouts", "FillArrays", "LinearAlgebra"] -git-tree-sha1 = "54cd829dd26330c42e1cf9df68470dd4df602c61" +git-tree-sha1 = "291532989f81db780e435452ccb2a5f902ff665f" uuid = "8e7c35d0-a365-5155-bbbb-fb81a777f24e" -version = "0.16.38" +version = "1.7.0" +weakdeps = ["Adapt", "BandedMatrices"] + + [deps.BlockArrays.extensions] + BlockArraysAdaptExt = "Adapt" + BlockArraysBandedMatricesExt = "BandedMatrices" [[deps.BlockBandedMatrices]] deps = ["ArrayLayouts", "BandedMatrices", "BlockArrays", "FillArrays", "LinearAlgebra", "MatrixFactorizations"] -git-tree-sha1 = "ee30b97bcee87889485293721b38c7f102bdd371" +git-tree-sha1 = "4eef2d2793002ef8221fe561cc822eb252afa72f" uuid = "ffab5731-97b5-5995-9138-79e8c1846df0" -version = "0.12.5" +version = "0.13.4" weakdeps = ["SparseArrays"] [deps.BlockBandedMatrices.extensions] BlockBandedMatricesSparseArraysExt = "SparseArrays" +[[deps.BracketingNonlinearSolve]] +deps = ["CommonSolve", "ConcreteStructs", "NonlinearSolveBase", "PrecompileTools", "Reexport", "SciMLBase"] +git-tree-sha1 = "a9014924595b7a2c1dd14aac516e38fa10ada656" +uuid = "70df07ce-3d50-431d-a3e7-ca6ddb60ac1e" +version = "1.3.0" +weakdeps = ["ChainRulesCore", "ForwardDiff"] + + [deps.BracketingNonlinearSolve.extensions] + BracketingNonlinearSolveChainRulesCoreExt = ["ChainRulesCore", "ForwardDiff"] + BracketingNonlinearSolveForwardDiffExt = "ForwardDiff" + [[deps.Bzip2_jll]] -deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] -git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2" +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "1b96ea4a01afe0ea4090c5c8039690672dd13f2e" uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0" -version = "1.0.8+0" +version = "1.0.9+0" [[deps.CEnum]] -git-tree-sha1 = "eb4cb44a499229b3b8426dcfb5dd85333951ff90" +git-tree-sha1 = "389ad5c84de1ae7cf0e28e381131c98ea87d54fc" uuid = "fa961155-64e5-5f13-b03f-caf6b980ea82" -version = "0.4.2" +version = "0.5.0" [[deps.CPUSummary]] deps = ["CpuId", "IfElse", "PrecompileTools", "Static"] -git-tree-sha1 = "89e0654ed8c7aebad6d5ad235d6242c2d737a928" +git-tree-sha1 = "5a97e67919535d6841172016c9530fd69494e5ec" uuid = "2a0fbf3d-bb9c-48f3-b0a9-814d99fd7ab9" -version = "0.2.3" - -[[deps.CSTParser]] -deps = ["Tokenize"] -git-tree-sha1 = "3ddd48d200eb8ddf9cb3e0189fc059fd49b97c1f" -uuid = "00ebfdb7-1f24-5e51-bd34-a7502290713f" -version = "3.3.6" +version = "0.2.6" [[deps.Cairo_jll]] -deps = ["Artifacts", "Bzip2_jll", "CompilerSupportLibraries_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"] -git-tree-sha1 = "4b859a208b2397a7a623a03449e4636bdb17bcf2" +deps = ["Artifacts", "Bzip2_jll", "CompilerSupportLibraries_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"] +git-tree-sha1 = 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"InfiniteArrays", "InfiniteLinearAlgebra", "IntervalSets", "LazyArrays", "LazyBandedMatrices", "LinearAlgebra", "QuasiArrays", "RecurrenceRelationshipArrays", "RecurrenceRelationships", "SpecialFunctions"] +git-tree-sha1 = "8829f73e9f99a1427f62d179210245f009378778" +uuid = "b30e2e7b-c4ee-47da-9d5f-2c5c27239acd" +version = "0.15.3" + + [deps.ClassicalOrthogonalPolynomials.extensions] + ClassicalOrthogonalPolynomialsMutableArithmeticsExt = "MutableArithmetics" + + [deps.ClassicalOrthogonalPolynomials.weakdeps] + MutableArithmetics = "d8a4904e-b15c-11e9-3269-09a3773c0cb0" [[deps.CloseOpenIntervals]] deps = ["Static", "StaticArrayInterface"] -git-tree-sha1 = "70232f82ffaab9dc52585e0dd043b5e0c6b714f1" +git-tree-sha1 = "05ba0d07cd4fd8b7a39541e31a7b0254704ea581" uuid = "fb6a15b2-703c-40df-9091-08a04967cfa9" -version = "0.1.12" +version = "0.1.13" [[deps.CodecZlib]] deps = ["TranscodingStreams", "Zlib_jll"] -git-tree-sha1 = "02aa26a4cf76381be7f66e020a3eddeb27b0a092" +git-tree-sha1 = 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"TensorCore"] -git-tree-sha1 = "a1f44953f2382ebb937d60dafbe2deea4bd23249" +git-tree-sha1 = "8b3b6f87ce8f65a2b4f857528fd8d70086cd72b1" uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4" -version = "0.10.0" +version = "0.11.0" weakdeps = ["SpecialFunctions"] [deps.ColorVectorSpace.extensions] @@ -262,20 +278,9 @@ weakdeps = ["SpecialFunctions"] [[deps.Colors]] deps = ["ColorTypes", "FixedPointNumbers", "Reexport"] -git-tree-sha1 = "fc08e5930ee9a4e03f84bfb5211cb54e7769758a" +git-tree-sha1 = "37ea44092930b1811e666c3bc38065d7d87fcc74" uuid = "5ae59095-9a9b-59fe-a467-6f913c188581" -version = "0.12.10" - -[[deps.Combinatorics]] -git-tree-sha1 = "08c8b6831dc00bfea825826be0bc8336fc369860" -uuid = "861a8166-3701-5b0c-9a16-15d98fcdc6aa" -version = "1.0.2" - -[[deps.CommonMark]] -deps = ["Crayons", "JSON", "PrecompileTools", "URIs"] -git-tree-sha1 = "532c4185d3c9037c0237546d817858b23cf9e071" -uuid = "a80b9123-70ca-4bc0-993e-6e3bcb318db6" -version = "0.8.12" +version = "0.13.1" [[deps.CommonSolve]] git-tree-sha1 = "0eee5eb66b1cf62cd6ad1b460238e60e4b09400c" @@ -283,16 +288,21 @@ uuid = "38540f10-b2f7-11e9-35d8-d573e4eb0ff2" version = "0.2.4" [[deps.CommonSubexpressions]] -deps = ["MacroTools", "Test"] -git-tree-sha1 = "7b8a93dba8af7e3b42fecabf646260105ac373f7" +deps = ["MacroTools"] +git-tree-sha1 = "cda2cfaebb4be89c9084adaca7dd7333369715c5" uuid = "bbf7d656-a473-5ed7-a52c-81e309532950" -version = "0.3.0" +version = "0.3.1" + +[[deps.CommonWorldInvalidations]] +git-tree-sha1 = "ae52d1c52048455e85a387fbee9be553ec2b68d0" +uuid = "f70d9fcc-98c5-4d4a-abd7-e4cdeebd8ca8" +version = "1.0.0" [[deps.Compat]] -deps = ["UUIDs"] -git-tree-sha1 = "e460f044ca8b99be31d35fe54fc33a5c33dd8ed7" +deps = ["TOML", "UUIDs"] +git-tree-sha1 = "0037835448781bb46feb39866934e243886d756a" uuid = "34da2185-b29b-5c13-b0c7-acf172513d20" -version = "4.9.0" +version = "4.18.0" weakdeps = ["Dates", "LinearAlgebra"] [deps.Compat.extensions] @@ -301,40 +311,68 @@ weakdeps = ["Dates", "LinearAlgebra"] [[deps.CompilerSupportLibraries_jll]] deps = ["Artifacts", "Libdl"] uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae" -version = "1.0.2+0" +version = "1.1.1+0" [[deps.CompositeTypes]] -git-tree-sha1 = "02d2316b7ffceff992f3096ae48c7829a8aa0638" +git-tree-sha1 = "bce26c3dab336582805503bed209faab1c279768" uuid = "b152e2b5-7a66-4b01-a709-34e65c35f657" -version = "0.1.3" +version = "0.1.4" + +[[deps.CompositionsBase]] +git-tree-sha1 = "802bb88cd69dfd1509f6670416bd4434015693ad" +uuid = "a33af91c-f02d-484b-be07-31d278c5ca2b" +version = "0.1.2" +weakdeps = ["InverseFunctions"] + + [deps.CompositionsBase.extensions] + CompositionsBaseInverseFunctionsExt = "InverseFunctions" + +[[deps.ConcreteStructs]] +git-tree-sha1 = "f749037478283d372048690eb3b5f92a79432b34" +uuid = "2569d6c7-a4a2-43d3-a901-331e8e4be471" +version = "0.2.3" [[deps.ConcurrentUtilities]] deps = ["Serialization", "Sockets"] -git-tree-sha1 = "5372dbbf8f0bdb8c700db5367132925c0771ef7e" +git-tree-sha1 = 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"ArrayLayouts", "BandedMatrices", "BlockArrays", "DomainSets", "FillArrays", "InfiniteArrays", "Infinities", "IntervalSets", "LazyArrays", "LinearAlgebra", "QuasiArrays", "StaticArrays"] +git-tree-sha1 = "003c9419d0f7d59f2a1a02fcae95272b655aa2a0" +uuid = "7ae1f121-cc2c-504b-ac30-9b923412ae5c" +version = "0.19.3" + + [deps.ContinuumArrays.extensions] + ContinuumArraysMakieExt = "Makie" + ContinuumArraysRecipesBaseExt = "RecipesBase" + + [deps.ContinuumArrays.weakdeps] + Makie = "ee78f7c6-11fb-53f2-987a-cfe4a2b5a57a" + RecipesBase = "3cdcf5f2-1ef4-517c-9805-6587b60abb01" + [[deps.Contour]] -git-tree-sha1 = "d05d9e7b7aedff4e5b51a029dced05cfb6125781" +git-tree-sha1 = "439e35b0b36e2e5881738abc8857bd92ad6ff9a8" uuid = "d38c429a-6771-53c6-b99e-75d170b6e991" -version = "0.6.2" +version = "0.6.3" [[deps.CpuId]] deps = ["Markdown"] @@ -348,21 +386,25 @@ uuid = "a8cc5b0e-0ffa-5ad4-8c14-923d3ee1735f" version = "4.1.1" [[deps.DSP]] -deps = ["Compat", "FFTW", "IterTools", "LinearAlgebra", "Polynomials", "Random", "Reexport", "SpecialFunctions", "Statistics"] -git-tree-sha1 = "da8b06f89fce9996443010ef92572b193f8dca1f" +deps = ["Bessels", "FFTW", "IterTools", "LinearAlgebra", "Polynomials", "Random", "Reexport", "SpecialFunctions", "Statistics"] +git-tree-sha1 = "5989debfc3b38f736e69724818210c67ffee4352" uuid = "717857b8-e6f2-59f4-9121-6e50c889abd2" -version = "0.7.8" +version = "0.8.4" +weakdeps = ["OffsetArrays"] + + [deps.DSP.extensions] + OffsetArraysExt = "OffsetArrays" [[deps.DataAPI]] -git-tree-sha1 = "8da84edb865b0b5b0100c0666a9bc9a0b71c553c" +git-tree-sha1 = "abe83f3a2f1b857aac70ef8b269080af17764bbe" uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a" -version = "1.15.0" +version = "1.16.0" [[deps.DataStructures]] deps = ["Compat", "InteractiveUtils", "OrderedCollections"] -git-tree-sha1 = "3dbd312d370723b6bb43ba9d02fc36abade4518d" +git-tree-sha1 = "4e1fe97fdaed23e9dc21d4d664bea76b65fc50a0" uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8" -version = "0.18.15" +version = "0.18.22" [[deps.DataValueInterfaces]] git-tree-sha1 = "bfc1187b79289637fa0ef6d4436ebdfe6905cbd6" @@ -373,6 +415,12 @@ version = "1.0.0" deps = ["Printf"] uuid = "ade2ca70-3891-5945-98fb-dc099432e06a" +[[deps.Dbus_jll]] +deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl"] +git-tree-sha1 = "473e9afc9cf30814eb67ffa5f2db7df82c3ad9fd" +uuid = "ee1fde0b-3d02-5ea6-8484-8dfef6360eab" +version = "1.16.2+0" + [[deps.DelimitedFiles]] deps = ["Mmap"] git-tree-sha1 = "9e2f36d3c96a820c678f2f1f1782582fcf685bae" @@ -380,51 +428,56 @@ uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab" version = "1.9.1" [[deps.DiffEqBase]] -deps = ["ArrayInterface", "ChainRulesCore", "DataStructures", "DocStringExtensions", "EnumX", "FastBroadcast", "ForwardDiff", "FunctionWrappers", "FunctionWrappersWrappers", "LinearAlgebra", "Logging", "Markdown", "MuladdMacro", "Parameters", "PreallocationTools", "Printf", "RecursiveArrayTools", "Reexport", "Requires", "SciMLBase", "SciMLOperators", "Setfield", "SparseArrays", "Static", "StaticArraysCore", "Statistics", "Tricks", "TruncatedStacktraces", "ZygoteRules"] -git-tree-sha1 = "df8638dbfa03d1b336c410e23a9dfbf89cb53937" +deps = ["ArrayInterface", "ConcreteStructs", "DataStructures", "DocStringExtensions", "EnumX", "EnzymeCore", "FastBroadcast", "FastClosures", "FastPower", "FunctionWrappers", "FunctionWrappersWrappers", "LinearAlgebra", "Logging", "Markdown", "MuladdMacro", "Parameters", "PrecompileTools", "Printf", "RecursiveArrayTools", "Reexport", "SciMLBase", "SciMLOperators", "SciMLStructures", "Setfield", "Static", "StaticArraysCore", "Statistics", "SymbolicIndexingInterface", "TruncatedStacktraces"] +git-tree-sha1 = "6f8f25122389ba34e491bdfe143ae16fda37cad3" uuid = "2b5f629d-d688-5b77-993f-72d75c75574e" -version = "6.128.2" +version = "6.180.0" [deps.DiffEqBase.extensions] + DiffEqBaseCUDAExt = "CUDA" + DiffEqBaseChainRulesCoreExt = "ChainRulesCore" DiffEqBaseDistributionsExt = "Distributions" + DiffEqBaseEnzymeExt = ["ChainRulesCore", "Enzyme"] + DiffEqBaseForwardDiffExt = ["ForwardDiff"] + DiffEqBaseGTPSAExt = "GTPSA" DiffEqBaseGeneralizedGeneratedExt = "GeneralizedGenerated" DiffEqBaseMPIExt = "MPI" DiffEqBaseMeasurementsExt = "Measurements" DiffEqBaseMonteCarloMeasurementsExt = "MonteCarloMeasurements" + DiffEqBaseMooncakeExt = "Mooncake" DiffEqBaseReverseDiffExt = "ReverseDiff" + DiffEqBaseSparseArraysExt = "SparseArrays" DiffEqBaseTrackerExt = "Tracker" DiffEqBaseUnitfulExt = "Unitful" - DiffEqBaseZygoteExt = "Zygote" [deps.DiffEqBase.weakdeps] + CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba" + ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" + Enzyme = "7da242da-08ed-463a-9acd-ee780be4f1d9" + ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" + GTPSA = "b27dd330-f138-47c5-815b-40db9dd9b6e8" GeneralizedGenerated = "6b9d7cbe-bcb9-11e9-073f-15a7a543e2eb" MPI = "da04e1cc-30fd-572f-bb4f-1f8673147195" Measurements = "eff96d63-e80a-5855-80a2-b1b0885c5ab7" MonteCarloMeasurements = "0987c9cc-fe09-11e8-30f0-b96dd679fdca" + Mooncake = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6" ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" + SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c" Unitful = "1986cc42-f94f-5a68-af5c-568840ba703d" - Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" - -[[deps.DiffEqCallbacks]] -deps = ["DataStructures", "DiffEqBase", "ForwardDiff", "LinearAlgebra", "Markdown", "NLsolve", "Parameters", "RecipesBase", "RecursiveArrayTools", "SciMLBase", "StaticArraysCore"] -git-tree-sha1 = "9c7d3a84264d935f6981504388b202a770113faa" -uuid = "459566f4-90b8-5000-8ac3-15dfb0a30def" -version = "2.29.1" -weakdeps = ["OrdinaryDiffEq", "Sundials"] [[deps.DiffEqDevTools]] -deps = ["DiffEqBase", "DiffEqNoiseProcess", "Distributed", "LinearAlgebra", "Logging", "NLsolve", "RecipesBase", "RecursiveArrayTools", "RootedTrees", "SciMLBase", "Statistics"] -git-tree-sha1 = "9fe41ca501ce654f1876735206816dca7fcd2c88" +deps = ["DiffEqBase", "DiffEqNoiseProcess", "Distributed", "LinearAlgebra", "Logging", "NLsolve", "RecipesBase", "RecursiveArrayTools", "RootedTrees", "SciMLBase", "Statistics", "StructArrays"] +git-tree-sha1 = "53aa030c02c786270542debbb577b53f6383796b" uuid = "f3b72e0c-5b89-59e1-b016-84e28bfd966d" -version = "2.35.0" +version = "2.48.0" [[deps.DiffEqNoiseProcess]] -deps = ["DiffEqBase", "Distributions", "GPUArraysCore", "LinearAlgebra", "Markdown", "Optim", "PoissonRandom", "QuadGK", "Random", "Random123", "RandomNumbers", "RecipesBase", "RecursiveArrayTools", "Requires", "ResettableStacks", "SciMLBase", "StaticArraysCore", "Statistics"] -git-tree-sha1 = "6b02e9c9d0d4cacf2b20f36c33710b8b415c5194" +deps = ["DiffEqBase", "Distributions", "GPUArraysCore", "LinearAlgebra", "Markdown", "Optim", "PoissonRandom", "QuadGK", "Random", "Random123", "RandomNumbers", "RecipesBase", "RecursiveArrayTools", "ResettableStacks", "SciMLBase", "StaticArraysCore", "Statistics"] +git-tree-sha1 = "516d553f5deee7c55b2945b5edf05b6542837887" uuid = "77a26b50-5914-5dd7-bc55-306e6241c503" -version = "5.18.0" +version = "5.24.1" [deps.DiffEqNoiseProcess.extensions] DiffEqNoiseProcessReverseDiffExt = "ReverseDiff" @@ -444,14 +497,65 @@ git-tree-sha1 = "23163d55f885173722d1e4cf0f6110cdbaf7e272" uuid = "b552c78f-8df3-52c6-915a-8e097449b14b" version = "1.15.1" +[[deps.DifferentiationInterface]] +deps = ["ADTypes", "LinearAlgebra"] +git-tree-sha1 = "f620da805b82bec64ab4d5f881c7592c82dbc08a" +uuid = "a0c0ee7d-e4b9-4e03-894e-1c5f64a51d63" +version = "0.7.3" + + [deps.DifferentiationInterface.extensions] + DifferentiationInterfaceChainRulesCoreExt = "ChainRulesCore" + DifferentiationInterfaceDiffractorExt = "Diffractor" + DifferentiationInterfaceEnzymeExt = ["EnzymeCore", "Enzyme"] + DifferentiationInterfaceFastDifferentiationExt = "FastDifferentiation" + DifferentiationInterfaceFiniteDiffExt = "FiniteDiff" + DifferentiationInterfaceFiniteDifferencesExt = "FiniteDifferences" + DifferentiationInterfaceForwardDiffExt = ["ForwardDiff", "DiffResults"] + DifferentiationInterfaceGPUArraysCoreExt = "GPUArraysCore" + DifferentiationInterfaceGTPSAExt = "GTPSA" + DifferentiationInterfaceMooncakeExt = "Mooncake" + DifferentiationInterfacePolyesterForwardDiffExt = ["PolyesterForwardDiff", "ForwardDiff", "DiffResults"] + DifferentiationInterfaceReverseDiffExt = ["ReverseDiff", "DiffResults"] + DifferentiationInterfaceSparseArraysExt = "SparseArrays" + DifferentiationInterfaceSparseConnectivityTracerExt = "SparseConnectivityTracer" + DifferentiationInterfaceSparseMatrixColoringsExt = "SparseMatrixColorings" + DifferentiationInterfaceStaticArraysExt = "StaticArrays" + DifferentiationInterfaceSymbolicsExt = "Symbolics" + DifferentiationInterfaceTrackerExt = "Tracker" + DifferentiationInterfaceZygoteExt = ["Zygote", "ForwardDiff"] + + [deps.DifferentiationInterface.weakdeps] + ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" + DiffResults = "163ba53b-c6d8-5494-b064-1a9d43ac40c5" + Diffractor = "9f5e2b26-1114-432f-b630-d3fe2085c51c" + Enzyme = "7da242da-08ed-463a-9acd-ee780be4f1d9" + EnzymeCore = "f151be2c-9106-41f4-ab19-57ee4f262869" + FastDifferentiation = "eb9bf01b-bf85-4b60-bf87-ee5de06c00be" + FiniteDiff = "6a86dc24-6348-571c-b903-95158fe2bd41" + FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000" + ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" + GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527" + GTPSA = "b27dd330-f138-47c5-815b-40db9dd9b6e8" + Mooncake = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6" + PolyesterForwardDiff = "98d1487c-24ca-40b6-b7ab-df2af84e126b" + ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" + SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" + SparseConnectivityTracer = "9f842d2f-2579-4b1d-911e-f412cf18a3f5" + SparseMatrixColorings = "0a514795-09f3-496d-8182-132a7b665d35" + StaticArrays = "90137ffa-7385-5640-81b9-e52037218182" + Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7" + Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c" + Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" + [[deps.Distances]] deps = ["LinearAlgebra", "Statistics", "StatsAPI"] -git-tree-sha1 = "b6def76ffad15143924a2199f72a5cd883a2e8a9" +git-tree-sha1 = "c7e3a542b999843086e2f29dac96a618c105be1d" uuid = "b4f34e82-e78d-54a5-968a-f98e89d6e8f7" -version = "0.10.9" -weakdeps = ["SparseArrays"] +version = "0.10.12" +weakdeps = ["ChainRulesCore", "SparseArrays"] [deps.Distances.extensions] + DistancesChainRulesCoreExt = "ChainRulesCore" DistancesSparseArraysExt = "SparseArrays" [[deps.Distributed]] @@ -459,105 +563,126 @@ deps = ["Random", "Serialization", "Sockets"] uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b" [[deps.Distributions]] -deps = ["FillArrays", "LinearAlgebra", "PDMats", "Printf", "QuadGK", "Random", "SpecialFunctions", "Statistics", "StatsAPI", "StatsBase", "StatsFuns", "Test"] -git-tree-sha1 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= ["Pardiso", "SparseArrays"] + LinearSolveRecursiveFactorizationExt = "RecursiveFactorization" + LinearSolveSparseArraysExt = "SparseArrays" + LinearSolveSparspakExt = ["SparseArrays", "Sparspak"] [deps.LinearSolve.weakdeps] + BandedMatrices = "aae01518-5342-5314-be14-df237901396f" + BlockDiagonals = "0a1fb500-61f7-11e9-3c65-f5ef3456f9f0" CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba" + CUDSS = "45b445bb-4962-46a0-9369-b4df9d0f772e" + EnzymeCore = "f151be2c-9106-41f4-ab19-57ee4f262869" + FastAlmostBandedMatrices = "9d29842c-ecb8-4973-b1e9-a27b1157504e" + FastLapackInterface = "29a986be-02c6-4525-aec4-84b980013641" + ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" HYPRE = "b5ffcf37-a2bd-41ab-a3da-4bd9bc8ad771" IterativeSolvers = "42fd0dbc-a981-5370-80f2-aaf504508153" + KernelAbstractions = "63c18a36-062a-441e-b654-da1e3ab1ce7c" KrylovKit = "0b1a1467-8014-51b9-945f-bf0ae24f4b77" - MKL_jll = "856f044c-d86e-5d09-b602-aeab76dc8ba7" Metal = "dde4c033-4e86-420c-a63e-0dd931031962" Pardiso = "46dd5b70-b6fb-5a00-ae2d-e8fea33afaf2" + RecursiveFactorization = "f2c3362d-daeb-58d1-803e-2bc74f2840b4" + SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" + Sparspak = "e56a9233-b9d6-4f03-8d0f-1825330902ac" [[deps.LogExpFunctions]] deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"] -git-tree-sha1 = "7d6dd4e9212aebaeed356de34ccf262a3cd415aa" +git-tree-sha1 = "13ca9e2586b89836fd20cccf56e57e2b9ae7f38f" uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688" -version = "0.3.26" +version = "0.3.29" [deps.LogExpFunctions.extensions] LogExpFunctionsChainRulesCoreExt = "ChainRulesCore" @@ -1150,52 +1276,36 @@ uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" [[deps.LoggingExtras]] deps = ["Dates", "Logging"] -git-tree-sha1 = "0d097476b6c381ab7906460ef1ef1638fbce1d91" +git-tree-sha1 = "f02b56007b064fbfddb4c9cd60161b6dd0f40df3" uuid = "e6f89c97-d47a-5376-807f-9c37f3926c36" -version = "1.0.2" +version = "1.1.0" [[deps.LoopVectorization]] -deps = ["ArrayInterface", 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[deps.RecursiveArrayTools.extensions] + RecursiveArrayToolsFastBroadcastExt = "FastBroadcast" + RecursiveArrayToolsForwardDiffExt = "ForwardDiff" + RecursiveArrayToolsKernelAbstractionsExt = "KernelAbstractions" RecursiveArrayToolsMeasurementsExt = "Measurements" + RecursiveArrayToolsMonteCarloMeasurementsExt = "MonteCarloMeasurements" + RecursiveArrayToolsReverseDiffExt = ["ReverseDiff", "Zygote"] + RecursiveArrayToolsSparseArraysExt = ["SparseArrays"] + RecursiveArrayToolsStructArraysExt = "StructArrays" RecursiveArrayToolsTrackerExt = "Tracker" RecursiveArrayToolsZygoteExt = "Zygote" [deps.RecursiveArrayTools.weakdeps] + FastBroadcast = "7034ab61-46d4-4ed7-9d0f-46aef9175898" + ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" + KernelAbstractions = "63c18a36-062a-441e-b654-da1e3ab1ce7c" Measurements = "eff96d63-e80a-5855-80a2-b1b0885c5ab7" + MonteCarloMeasurements = "0987c9cc-fe09-11e8-30f0-b96dd679fdca" + ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" + SparseArrays = 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+git-tree-sha1 = "62389eeff14780bfe55195b7204c0d8738436d64" uuid = "ae029012-a4dd-5104-9daa-d747884805df" -version = "1.3.0" +version = "1.3.1" [[deps.ResettableStacks]] deps = ["StaticArrays"] @@ -1661,21 +2055,21 @@ version = "1.1.1" [[deps.Rmath]] deps = ["Random", "Rmath_jll"] -git-tree-sha1 = "f65dcb5fa46aee0cf9ed6274ccbd597adc49aa7b" +git-tree-sha1 = "852bd0f55565a9e973fcfee83a84413270224dc4" uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa" -version = "0.7.1" +version = "0.8.0" [[deps.Rmath_jll]] -deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] -git-tree-sha1 = "6ed52fdd3382cf21947b15e8870ac0ddbff736da" +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "58cdd8fb2201a6267e1db87ff148dd6c1dbd8ad8" uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f" -version = "0.4.0+0" +version = "0.5.1+0" [[deps.RootedTrees]] -deps = ["Latexify", "LinearAlgebra", "Preferences", "RecipesBase", "Requires"] -git-tree-sha1 = "dc172b0f145b0921a25b9cd48a56559f72170f10" +deps = ["LaTeXStrings", "Latexify", "LinearAlgebra", "Preferences", "RecipesBase", "Requires"] +git-tree-sha1 = "c0c464d3063e46e4128d21fd677ca575ace44fdc" uuid = "47965b36-3f3e-11e9-0dcf-4570dfd42a8c" -version = "2.19.2" +version = "2.23.1" weakdeps = ["Plots"] [deps.RootedTrees.extensions] @@ -1683,20 +2077,14 @@ weakdeps = ["Plots"] [[deps.RuntimeGeneratedFunctions]] deps = ["ExprTools", "SHA", "Serialization"] -git-tree-sha1 = "6aacc5eefe8415f47b3e34214c1d79d2674a0ba2" +git-tree-sha1 = "86a8a8b783481e1ea6b9c91dd949cb32191f8ab4" uuid = "7e49a35a-f44a-4d26-94aa-eba1b4ca6b47" -version = "0.5.12" +version = "0.5.15" [[deps.SHA]] uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" version = "0.7.0" -[[deps.SIMD]] -deps = ["PrecompileTools"] -git-tree-sha1 = "0e270732477b9e551d884e6b07e23bb2ec947790" -uuid = "fdea26ae-647d-5447-a871-4b548cad5224" -version = "3.4.5" - [[deps.SIMDTypes]] git-tree-sha1 = "330289636fb8107c5f32088d2741e9fd7a061a5c" uuid = "94e857df-77ce-4151-89e5-788b33177be4" @@ -1704,20 +2092,35 @@ 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"Xorg_xkeyboard_config_jll"] -git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1" uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" -version = "1.4.1+0" +version = "1.9.2+0" diff --git a/benchmarks/SimpleHandwrittenPDE/Project.toml b/benchmarks/SimpleHandwrittenPDE/Project.toml index 44f95075a..872555fc8 100644 --- a/benchmarks/SimpleHandwrittenPDE/Project.toml +++ b/benchmarks/SimpleHandwrittenPDE/Project.toml @@ -1,30 +1,20 @@ [deps] -ApproxFun = "28f2ccd6-bb30-5033-b560-165f7b14dc2f" +ClassicalOrthogonalPolynomials = "b30e2e7b-c4ee-47da-9d5f-2c5c27239acd" DiffEqDevTools = "f3b72e0c-5b89-59e1-b016-84e28bfd966d" -DomainSets = "5b8099bc-c8ec-5219-889f-1d9e522a28bf" LSODA = "7f56f5a3-f504-529b-bc02-0b1fe5e64312" LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" LinearSolve = "7ed4a6bd-45f5-4d41-b270-4a48e9bafcae" -MethodOfLines = "94925ecb-adb7-4558-8ed8-f975c56a0bf4" -ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" -ODEInterfaceDiffEq = "09606e27-ecf5-54fc-bb29-004bd9f985bf" OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" -RecursiveFactorization = "f2c3362d-daeb-58d1-803e-2bc74f2840b4" SciMLBenchmarks = "31c91b34-3c75-11e9-0341-95557aab0344" +SciMLOperators = "c0aeaf25-5076-4817-a8d5-81caf7dfa961" +SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" +SummationByPartsOperators = "9f78cca6-572e-554e-b819-917d2f1cf240" Sundials = "c3572dad-4567-51f8-b174-8c6c989267f4" [compat] -ApproxFun = "0.13" DiffEqDevTools = "2.22" -DomainSets = "0.6, 0.7" LSODA = "0.6, 0.7" -LinearSolve = "2, 3" -MethodOfLines = "0.9" -ModelingToolkit = "8, 10" -ODEInterfaceDiffEq = "3.7" -OrdinaryDiffEq = "5.41, 6" Plots = "1.4" -RecursiveFactorization = "0.2.23" SciMLBenchmarks = "0.1" Sundials = "4.2" diff --git a/benchmarks/SimpleHandwrittenPDE/allen_cahn_fdm_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/allen_cahn_fdm_wpd.jmd index e4fe891e6..33b124544 100644 --- a/benchmarks/SimpleHandwrittenPDE/allen_cahn_fdm_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/allen_cahn_fdm_wpd.jmd @@ -1,238 +1,271 @@ --- -title: Allen_Cahn FDM Work-Precision Diagrams -author: HAO HAO +title: Allen-Cahn Finite-Difference Method Work-Precision Diagrams +author: Arjit Seth --- -```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve -using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +## Problem Description + +The Allen-Cahn partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 2,~T = 1$, with the following initial and Dirichlet boundary conditions: +```math +\begin{align} + \partial_t u(t,x) & = p(u(t,x) - u(t,x)^3) + \epsilon \partial_x^2 u(t,x), \\ + u(0,x) & = \cos(2\pi x), \\ + u(t,-L) & = u(t,L) = 0. +\end{align} +``` +The spatial derivative operators are represented via finite difference approximations on an equispaced grid of points $x_s \in [-L, L]$. The coefficients $p = 3,~\epsilon = 10^{-3}$ are chosen to produce `interesting' behavior as seen in the reference solution below. +```math +\begin{align} + \frac{du}{dt} & = p(u - u^3) + \epsilon D_x^2 u, \\ + u(0,x_s) & = \cos(2\pi x_s), \\ + u(t,-L) & = u(t,L) = 0. +\end{align} ``` -Here is the Burgers equation using FDM. +### Implementation ```julia -# Define the linear and nonlinear terms -function lin_term(N) - dx = 1/(N + 1) - du = 1/2 * ones(N - 1) # super diagonal - dl = -1/2 * ones(N - 1) # lower diagonal - DiffEqArrayOperator(0.01 * (1/dx) * diagm(-1 => dl, 1 => du)) -end +using OrdinaryDiffEq +using DiffEqDevTools +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using SummationByPartsOperators +const SBP = SummationByPartsOperators +using Plots +gr(); +``` -function nl_term(N) - function (du, u, p, t) - du .= u .- u .^ 3 - end +```julia +# Nonlinear term: u - u^3 +function forcing_term(du, u, p, t) + du .= p * @. (u - u^3) + # Apply boundary conditions (ghost cells) + du[1] = 0.0 + du[end] = 0.0 end # Construct the problem -function allen_cahn(N) - f1 = lin_term(N) - f2 = nl_term(N) - dx = 1 / (N + 1) - xs = (1:N) * dx - - f0 = x -> 0.53*x + 0.47*sin(-1.5*pi*x) - x - u0 = f0.(xs) - prob = SplitODEProblem(f1, f2, u0, (0.0, 1.0)) - xs, prob +function allen_cahn(N, L) + # Linear and nonlinear terms + eps = 1e-3 # Diffusion coefficient + D2 = derivative_operator(MattssonSvärdNordström2004(); + derivative_order = 2, accuracy_order = 2, + xmin = -L, xmax = L, N = N) + + x = LinRange(-L, L, N) # Domain discretization + u0 = @. cos(2π * x) # Initial condition + p = 3.0 # Nonlinear coefficient + + prob = SplitODEProblem(MatrixOperator(eps * sparse(D2)), forcing_term, u0, (0.0, 1.0), p) + + x, prob end; ``` -Reference solution using Vern9 is below: +### Reference Solution ```julia -xs, prob = allen_cahn(100) -sol = solve(prob, RadauIIA5(autodiff = false); abstol = 1e-14, - reltol = 1e-14, dt = 1e-4, adaptive = false) -test_sol = TestSolution(sol); - -tslices = [0.0 0.25 0.50 0.75 1.0] -ys = hcat((sol(t) for t in tslices)...) -labels = ["t = $t" for t in tslices] -plot(xs, ys, label = labels) -``` +N = 128 # Number of grid points +L = 2.0 # Domain length +xs, prob = allen_cahn(N, L) +@time sol = solve(prob, AutoVern7(RadauIIA5(autodiff=false)); dt=1e-2, abstol=1e-14, reltol=1e-14, adaptive=true) -Linear solvers +test_sol = TestSolution(sol) # Reference solution for error estimation -```julia -const LS_Dense = LinSolveFactorize(lu) +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +plt = heatmap(xs, tslices, ys', xlabel="x", ylabel="t") ``` -## High tolerances +## Work-Precision Diagrams -## In-family comparisons +### High Tolerances -1.IMEX methods (dense linear solver) +#### Implicit-Explicit Methods ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(), :dts => 1e-3 * multipliers), +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), - Dict(:alg => SBDF2(), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] -@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, low order") -``` - -1.IMEX methods (Krylov linear solver) - -```julia -abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much -reltols = 0.1 .^ (1:4) -multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = KrylovJL_GMRES()), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers), - Dict(:alg => CNLF2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers), - Dict(:alg => SBDF2(linsolve = KrylovJL_GMRES()), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, Krylov linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` - 2. ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 5), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 20), :dts => 1e-3 * multipliers), +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler (caching)", "NorsettEuler (m=5)", "NorsettEuler (m=20)", - "ETDRK2 (caching)", "ETDRK2 (m=5)", "ETDRK2 (m=20)") + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-3 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-3 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, low order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers)] -labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"] +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => Tsit5(), :dts => 1e-5 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (caching)", + "Tsit5", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, low orders") +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances -## In-family comparisons +### Low Tolerances -1.IMEX methods (dense linear solver) +#### Implicit-Explicit Methods + +Dense/banded linear solvers. ```julia abstols = 0.1 .^ (7:13) reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3()), +setups = [ + Dict(:alg => KenCarp3()), Dict(:alg => KenCarp4()), Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense))] -labels = hcat("KenCarp3", "KenCarp4", "KenCarp5", "ARKODE3", "ARKODE4", "ARKODE5") + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, medium order") +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Band Linsolve, Low Tolerances") ``` -1.IMEX methods (Krylov linear solver) - +Krylov linear solvers. ```julia abstols = 0.1 .^ (7:13) reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES))] -labels = ["KenCarp3" "KenCarp4" "KenCarp5" "ARKODE3" "ARKODE4" "ARKODE5"] +setups = [ + Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, medium order") +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers), Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK3 (m=5)", "ETDRK4 (caching)", - "ETDRK4 (m=5)", "HochOst4 (caching)", "HochOst4 (m=5)") +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, medium order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense)), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES)), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat( - "KenCarp5 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp5 (Krylov linsolve)", - "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)") +setups = [ + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), +] +labels = hcat("ARKODE (nondiagonal linsolve)", "ETDRK3 ()", "ETDRK4 ()") @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, medium order") +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) diff --git a/benchmarks/SimpleHandwrittenPDE/allen_cahn_spectral_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/allen_cahn_spectral_wpd.jmd index cceacff91..bf65aff1c 100644 --- a/benchmarks/SimpleHandwrittenPDE/allen_cahn_spectral_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/allen_cahn_spectral_wpd.jmd @@ -1,231 +1,282 @@ --- title: Allen-Cahn Pseudospectral Methods Work-Precision Diagrams -author: HAO HAO +author: Arjit Seth --- + +## Problem Description + +The Allen-Cahn partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 1,~T = 1$, with the following initial and boundary conditions: +```math +\begin{aligned} + \partial_t u(t,x) & = u(t,x) - u^3(t,x) + \epsilon \partial_x^2 u(t,x), \\ + u(0,x) & = \cos(2\pi x), \\ + u(t,-L) & = u(t,L) = 1. +\end{aligned} +``` +The spatial derivative operators are represented via Chebyshev pseudospectral approximations. Here, the domain is discretized by projecting on a cosine-spaced grid of points $x_s \in [-1, 1]$; the solution is approximated on this grid via linear combinations of Chebyshev polynomial basis functions in space. The coefficient $p = 3,~\epsilon = 10^{-3}$ is chosen to produce `interesting' behavior as seen in the reference solution below. +```math +\begin{aligned} + \frac{du}{dt} & = p(u(t,x) - u^3(t,x)) + \epsilon D_x^2 u(t,x), \\ + u(0,x_s) & = \cos(2\pi x_s), \\ + u(t,-L) & = u(t,L) = 1. +\end{aligned} +``` + +### Implementation + ```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve +using OrdinaryDiffEq using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using ClassicalOrthogonalPolynomials +using Plots +gr(); ``` -Here is the Allen-Cahn equation using Chebyshev spectral methods. - ```julia -function cheb(N) - N==0 && return (0, 1) - x = cos.(pi*(0:N)/N) - c = [2; ones(N-1, 1); 2] .* (-1) .^ (0:N) - X = hcat([x for i in 1:(N + 1)]...) - dX = X-X' - D = (c*(1 ./ c)') ./ (dX+I) # off-diagonal entries - D = D .- Diagonal(vec(sum(D, dims = 2))) # diagonal entries - D, x -end -N = 128 -ChebD2, x = cheb(N) -xx = x -x = x[2:N] -w = 0.53*x + 0.47*sin.(-1.5*pi*x) - x # use w = u-x to make BCs homogeneous -u = [1; w+x; -1] - -ϵ=0.01 -D2=ϵ*(ChebD2 ^ 2)[2:N, 2:N] -function allen_cahn(du, u, x, t) - @. du = (u + x) - (u + x)^3 +# Nonlinear component of vector field +function f_nonlinear!(du, u, p, t) + M, D0, alpha, tmp = p + Tu = D0 * u # Pseudo-spectral representation of solution + @. tmp[2:end-1] = alpha * (Tu - Tu^3) + ldiv!(du, M, tmp) # Solve the linear system M * du = tmp end + +function allen_cahn(n, eps) + T = ChebyshevT() # Chebyshev basis + x1 = reverse(ChebyshevGrid{1}(n - 2)) # 1st kind points, sorted + x2 = reverse(ChebyshevGrid{2}(n)) # 2nd kind points, sorted + + V = T[x2, 1:n] # Vandermonde matrix, its inverse is transform from values to coefficients + D0 = diff(T, 0)[x1, 1:n] / V # discretisation of identity matrix + D2 = diff(T, 2)[x1, 1:n] / V # 2nd derivative from x2 to x1 + + B_l = [1; zeros(n-1)]' # Left Dirichlet conditions + B_r = [zeros(n-1); 1]' # Right Dirichlet + M = lu([B_l; D0; B_r]) # Mass matrix for the problem + u0 = cos.(2π * x2) # Initial condition + alpha = 3.0 # Time scaling factor + D2_bc = [zeros(1, n); D2; zeros(1, n)] # 2nd derivative with Dirichlet conditions + + # Problem setup + prob = SplitODEProblem( + MatrixOperator(eps * (M \ D2_bc)), # Linear operator with mass matrix inversion + f_nonlinear!, + u0, (0.0, 1.0), + (M, D0, alpha, similar(u0)) + ) + + return x2, prob +end; ``` -Reference solution using RadauIIA5 is below: +### Reference Solution ```julia -prob = SplitODEProblem(DiffEqArrayOperator(D2), allen_cahn, w, (0.0, 5.0), x) -sol = solve(prob, RadauIIA5(autodiff = false); reltol = 1e-14, abstol = 1e-14) -test_sol = TestSolution(sol) - -tslices=[0.0 1.0 2.0 3.0 5.0] -ys=hcat(([1; x .+ sol(t); -1] for t in tslices)...) -labels=["t=$t" for t in tslices] -plot(xx, ys, label = labels) -``` - -## High tolerances +N = 128 +eps = 1e-3 +xs, prob = allen_cahn(N, eps); -## In-family comparisons +@time sol = solve(prob, Rodas5(autodiff=false); dt=1e-4, reltol=1e-12, abstol=1e-12); -1.IMEX methods (dense linear solver) +test_sol = TestSolution(sol); # Reference solution for error estimation -```julia -abstols = 0.1 .^ (5:8) -reltols = 0.1 .^ (1:4) -multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(), :dts => 5e-3 * multipliers), - Dict(:alg => CNLF2(), :dts => 5e-4 * multipliers), - Dict(:alg => SBDF2(), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] -@time wp1 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, seconds = 5, - save_everystop = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp1, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, low order") +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +heatmap(xs, tslices, ys', xlabel="x", ylabel="t") ``` -1.IMEX methods (Krylov linear solver) +## Work-Precision Diagrams + +### High Tolerances + +#### Implicit-Explicit Methods ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = KrylovJL_GMRES()), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 5e-3 * multipliers), - Dict(:alg => CNLF2(linsolve = KrylovJL_GMRES()), :dts => 5e-4 * multipliers), - Dict(:alg => SBDF2(linsolve = KrylovJL_GMRES()), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] -@time wp1 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp1, label = labels, markershape = :auto, title = "IMEX methods, Krylov linsolve, low order") +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` - 2. ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 5), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 20), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 5), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler (caching)", "NorsettEuler (m=5)", "NorsettEuler (m=20)", - "ETDRK2 (caching)", "ETDRK2 (m=5)", "ETDRK2 (m=20)") -@time wp2 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp2, label = labels, markershape = :auto, title = "ExpRK methods, low order") +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` -## Between family comparisons + +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(), :dts => 5e-3 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 5e-3 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"] -@time wp3 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp3, label = labels, markershape = :auto, title = "Between family, low orders") +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (m=5)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances -## In-family comparisons +### Low Tolerances + -1.IMEX methods (dense/band linear solver) +#### Implicit-Explicit Methods + +Dense/banded linear solvers. ```julia abstols = 0.1 .^ (7:13) reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3()), +setups = [ + Dict(:alg => KenCarp3()), Dict(:alg => KenCarp4()), Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1))] -labels = hcat("KenCarp3", "KenCarp4", "KenCarp5", - "ARKODE3", "ARKODE4", "ARKODE5") -@time wp4 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp4, label = labels, markershape = :auto, title = "IMEX methods, dense/band linsolve, medium order") + #Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Band, jac_upper=1, jac_lower=1)), + #Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Band, jac_upper=1, jac_lower=1)), + #Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)) +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + #"ARKODE3", + #"ARKODE4", + #"ARKODE5", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Band Linsolve, Low Tolerances") ``` -1.IMEX methods (krylov linear solver) - +Krylov linear solvers. ```julia abstols = 0.1 .^ (7:13) reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES))] -labels = ["KenCarp3" "KenCarp4" "KenCarp5" "ARKODE3" "ARKODE4" "ARKODE5"] -@time wp4 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp4, label = labels, markershape = :auto, title = "IMEX methods, Krylov linsolve, medium order") +setups = [ + Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers), Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK3 (m=5)", "ETDRK4 (caching)", - "ETDRK4 (m=5)", "HochOst4 (caching)", "HochOst4 (m=5)") -@time wp5 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp5, label = labels, markershape = :auto, title = "ExpRK methods, medium order") +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense)), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES)), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat( - "KenCarp5 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp5 (Krylov linsolve)", - "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)") -@time wp6 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp6, label = labels, markershape = :auto, title = "Between family, medium order") +setups = [ + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers) +] +labels = hcat("ARKODE (Krylov linsolve)", "ETDRK3 ()", "ETDRK4 ()") +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) diff --git a/benchmarks/SimpleHandwrittenPDE/burgers_fdm_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/burgers_fdm_wpd.jmd index 0ad92b853..bc3b07b85 100644 --- a/benchmarks/SimpleHandwrittenPDE/burgers_fdm_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/burgers_fdm_wpd.jmd @@ -1,239 +1,275 @@ --- -title: Burgers FDM Work-Precision Diagrams -author: HAO HAO +title: Burgers' Equation Finite-Difference Method Work-Precision Diagrams +author: Arjit Seth --- +## Problem Description + +The Burgers' partial differential equation is solved on the domain $[0, L] \times [0, T] \in \mathbb R \times R,~L = 1,~T = 1$, with the following initial and boundary conditions: +```math +\begin{aligned} + \partial_t u(t,x) & = -\partial_x u^2(t,x) + \nu \partial_x^2 u(t,x), \\ + u(0,x) & = \exp\left[\frac{(x - 0.5)^2}{2 \times 0.05^2}\right], \\ + u(-L,x) & = u(L,x) = 0. +\end{aligned} +``` +The spatial derivative operators are represented via finite difference approximations on an equispaced grid of points $x_s \in [-L, L]$. The diffusion coefficient $\nu = 10^{-2}$ is chosen to produce `interesting' behavior as seen in the reference solution below. +```math +\begin{aligned} + \frac{du}{dt} & = -D_x u^2(t,x) + \nu D_x^2 u(t,x), \\ + u(0,x_s) & = \exp\left[\frac{(x_s - 0.5)^2}{2 \times 0.05^2}\right], \\ + u(t,-L) & = u(t,L) = 0. +\end{aligned} +``` + +### Implementation + ```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve +using OrdinaryDiffEq using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using SummationByPartsOperators +const SBP = SummationByPartsOperators +using Plots +gr(); ``` -Here is the Burgers equation using FDM. - ```julia -function lin_term(N, ϵ) - dx = 1/(N + 1) - d = -2 * ones(N) # main diagonal - du = ones(N - 1) # off diagonal - DiffEqArrayOperator((ϵ/dx^2) * diagm(-1 => du, 0 => d, 1 => du)) -end - -function nl_term(N) - dx = 1/(N + 1) - du = ones(N - 1) # super diagonal - dl = -ones(N - 1) # lower diagonal - D = (-1/(4*dx)) * diagm(-1 => dl, 1 => du) - - tmp = zeros(N) - function (du, u, p, t) - @. tmp = u^2 - mul!(du, D, tmp) - end -end - -# Construct the problem -function burgers(N, ϵ) - f1 = lin_term(N, ϵ) - f2 = nl_term(N) - dx = 1 / (N + 1) - xs = (1:N) * dx - - μ0 = 0.3; - σ0 = 0.05 - f0 = x -> exp(-(x - μ0)^2 / (2 * σ0^2)) - u0 = f0.(xs) - prob = SplitODEProblem(f1, f2, u0, (0.0, 1.0)) - xs, prob +nonlinear_convection!(du, u, p, t) = du .= (-p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2)) + +# Constructor +function burgers(N, L) + # Derivative operators + D1 = derivative_operator(MattssonSvärdNordström2004(); + derivative_order = 1, accuracy_order = 2, + xmin = -L, xmax = L, N = N) + D2 = derivative_operator(MattssonSvärdNordström2004(); + derivative_order = 2, accuracy_order = 2, + xmin = -L, xmax = L, N = N) + + x = LinRange(0, L, N) # Domain discretization + u0 = @. exp(-(x - 0.5)^2 / (2 * 0.05^2)) # Initial condition + nu = 1e-2 # Diffusion coefficient + alpha = 1.0 # Convection coefficient + p = (; D1, alpha) + + prob = SplitODEProblem(MatrixOperator(nu * sparse(D2)), nonlinear_convection!, u0, (0.0, 1.0), p) + + return x, prob end; ``` -Reference solution using Vern9 is below: +### Reference Solution + +Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision. ```julia -xs, prob = burgers(512, 1e-3) -sol = solve(prob, Vern9(); abstol = 1e-14, reltol = 1e-14) -test_sol = TestSolution(sol); - -tslices = [0.0 0.25 0.50 0.75 1.00] -ys = hcat((sol(t) for t in tslices)...) -labels = ["t = $t" for t in tslices] -plot(xs, ys, label = labels) +N = 128 +L = 1.0 +xs, prob = burgers(N, L) +sol = solve(prob, AutoVern7(Rodas5(autodiff=false)); abstol=1e-14, reltol=1e-14, adaptive=true) + +test_sol = TestSolution(sol); # Reference solution for error estimation + +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +p = heatmap(xs, tslices, ys', xlabel="x", ylabel="t") ``` -## High tolerances +## Work-Precision Diagrams -## In-family comparisons +### High Tolerances -1.IMEX methods (dense linear solver) +#### Implicit-Explicit Methods ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(), :dts => 1e-3 * multipliers), +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), - Dict(:alg => SBDF2(), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` -1.IMEX methods (Krylov linear solver) +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = KrylovJL_GMRES()), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers), - Dict(:alg => CNLF2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers), - Dict(:alg => SBDF2(linsolve = KrylovJL_GMRES()), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-3 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-3 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, Krylov linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` - 2. ExpRK methods + +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 5), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 20), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers)] -#Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-2 * multipliers)) matrix contains Inf or NaN -#Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-2 * multipliers) matrix contains Inf or NaN -labels = hcat("NorsettEuler (caching)", "NorsettEuler (m=5)", "NorsettEuler (m=20)", - "ETDRK2 (caching)")#, "ETDRK2 (m=5)"), "ETDRK2 (m=20)") +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, low order") +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Between family comparisons -```julia -abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much -reltols = 0.1 .^ (1:4) -multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers)] -labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"] -@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); +### Low Tolerances -plot(wp, label = labels, markershape = :auto, title = "Between family, low orders") -``` - -## Low tolerances -## In-family comparisons +#### Implicit-Explicit Methods -1.IMEX methods (dense linear solver) +Dense/banded linear solvers. ```julia -abstols = 0.1 .^ (8:13) -reltols = 0.1 .^ (5:10) -setups = [Dict(:alg => KenCarp3()), +abstols = 0.1 .^ (7:13) +reltols = 0.1 .^ (4:10) +setups = [ + Dict(:alg => KenCarp3()), Dict(:alg => KenCarp4()), Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense))] -labels = hcat("KenCarp3", "KenCarp4", "KenCarp5", "ARKODE3", "ARKODE4", "ARKODE5") + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)) +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, medium order") +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Band Linsolve, Low Tolerances") ``` -1.IMEX methods (Krylov linear solver) - +Krylov linear solvers. ```julia -abstols = 0.1 .^ (8:13) -reltols = 0.1 .^ (5:10) -setups = [Dict(:alg => KenCarp3(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES))] -labels = ["KenCarp3" "KenCarp4" "KenCarp5" "ARKODE3" "ARKODE4" "ARKODE5"] +abstols = 0.1 .^ (7:13) +reltols = 0.1 .^ (4:10) +setups = [ + Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, medium order") +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers), Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK3 (m=5)", "ETDRK4 (caching)", - "ETDRK4 (m=5)", "HochOst4 (caching)", "HochOst4 (m=5)") +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, medium order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia -abstols = 0.1 .^ (8:13) -reltols = 0.1 .^ (5:10) -multipliers = 0.5 .^ (0:5) -setups = [Dict(:alg => KenCarp4()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense)), - Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES)), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] +abstols = 0.1 .^ (7:11) +reltols = 0.1 .^ (4:8) +multipliers = 0.5 .^ (0:4) +setups = [ + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), +] labels = hcat( - "KenCarp4 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp4 (Krylov linsolve)", - "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)") + "ARKODE (Band linsolve)", + "ETDRK3 (caching)", + "ETDRK4 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5));#162s + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, medium order") +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) diff --git a/benchmarks/SimpleHandwrittenPDE/burgers_spectral_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/burgers_spectral_wpd.jmd index 1e83075fd..003fbfd15 100644 --- a/benchmarks/SimpleHandwrittenPDE/burgers_spectral_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/burgers_spectral_wpd.jmd @@ -1,187 +1,280 @@ --- -title: Burgers Pseudospectral Methods Work-Precision Diagrams -author: HAO HAO +title: Burgers' Equation Pseudospectral Method Work-Precision Diagrams +author: Arjit Seth --- +### Problem Description + +The Burgers' partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 1,~T = 1$, with the following initial and boundary conditions: +```math +\begin{aligned} + \partial_t u(t,x) & = -\partial_x u^2(t,x) + \nu \partial_x^2 u(t,x), \\ + u(0,x) & = \exp\left(\frac{x^2}{2 \times 0.01^2}\right), \\ + u(t,-L) & = u(t,L) = 0. +\end{aligned} +``` +The spatial derivative operators are represented via Chebyshev pseudospectral approximations. Here, the domain is discretized by projecting on a cosine-spaced grid of points $x_s \in [-1, 1]$; the solution is approximated on this grid via linear combinations of Chebyshev polynomial basis functions in space. The diffusion coefficient $\nu = 10^{-2}$ is chosen to produce `interesting' behavior as seen in the reference solution below. +```math +\begin{aligned} + \frac{du}{dt} & = -D_x u^2(t,x) + \nu D_x^2 u(t,x), \\ + u(0,x_s) & = \exp\left(\frac{x_s^2}{2 \times 0.01^2}\right), \\ + u(t,-L) & = u(t,L) = 0. +\end{aligned} +``` + +### Implementation + ```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve +using OrdinaryDiffEq using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using ClassicalOrthogonalPolynomials +const COP = ClassicalOrthogonalPolynomials +using Plots +gr(); ``` -Here is the Burgers equation using Fourier spectral methods. - ```julia -S = Fourier() -n = 512 -x = points(S, n) -D2 = Derivative(S, 2)[1:n, 1:n] -D = (Derivative(S) → S)[1:n, 1:n] -T = ApproxFun.plan_transform(S, n) -Ti = ApproxFun.plan_itransform(S, n) - -û₀ = T*cos.(cos.(x .- 0.1)) -A = 0.03*D2 -tmp = similar(û₀) -p = (D, D2, T, Ti, tmp, similar(tmp)) -function burgers_nl(dû, û, p, t) - D, D2, T, Ti, u, tmp = p - mul!(tmp, D, û) - mul!(u, Ti, tmp) - mul!(tmp, Ti, û) - @. tmp = tmp*u - mul!(u, T, tmp) - @. dû = - u +# Nonlinear component of vector field +function f_nonlinear!(du, u, p, t) + M, D0, D1, tmp = p + tmp[2:end-1] .= -(D0 * u) .* (D1 * u) # Nonlinear convection + ldiv!(du, M, tmp) # Solve the linear system M * du = tmp end -``` -Reference solution using Rodas5 is below: +# Constructor +function burgers(n) + T = ChebyshevT() # Chebyshev basis + x1 = reverse(ChebyshevGrid{1}(n - 2)) # 1st kind points, sorted + x2 = reverse(ChebyshevGrid{2}(n)) # 2nd kind points, sorted + V = T[x2, 1:n] # Vandermonde matrix, its inverse is transform from values to coefficients + D0 = diff(T, 0)[x1, 1:n] / V # 0th derivative from x2 to x1 (identity) + D1 = diff(T, 1)[x1, 1:n] / V # 1st derivative from x2 to x1 + D2 = diff(T, 2)[x1, 1:n] / V # 2nd derivative from x2 to x1 -```julia -prob = SplitODEProblem(DiffEqArrayOperator(Diagonal(A)), burgers_nl, û₀, (0.0, 5.0), p) -sol = solve(prob, Rodas5(autodiff = false); reltol = 1e-12, abstol = 1e-12) -test_sol = TestSolution(sol) - -tslices=[0.0 1.0 2.0 3.0 5.0] -ys=hcat((Ti*sol(t) for t in tslices)...) -labels=["t=$t" for t in tslices] -plot(x, ys, label = labels) + + u0 = @. exp(-x2^2 / (2 * 0.1^2)) # Initial condition + B_l = [1 zeros(1, n-1)] # Left Dirichlet conditions + B_r = reverse(B_l) # Right Dirichlet + M = lu([B_l; D0; B_r]) # Mass matrix for the problem + nu = 1e-2 # Diffusion coefficient + D2_bc = [zeros(1, n); D2; zeros(1, n)] # 2nd derivative with boundary conditions + prob = SplitODEProblem( + MatrixOperator(nu * (M \ D2_bc)), + f_nonlinear!, + u0, (0.0, 1.0), + (M, D0, D1, similar(u0)) + ) + + return x2, prob +end; ``` -## High tolerances +### Reference Solution ```julia -diag_linsolve=LinSolveFactorize(W->let tmp = tmp - for i in 1:size(W, 1) - tmp[i] = W[i, i] - end - Diagonal(tmp) -end) +n = 128 # Number of Chebyshev points +xs, prob = burgers(n) +@time sol = solve(prob, Rodas5(autodiff=false), reltol = 1e-12, abstol = 1e-12) + +test_sol = TestSolution(sol) # Reference solution for error estimation + +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +heatmap(xs, tslices, ys') ``` -## In-family comparisons +## Work-Precision Diagrams + +### High Tolerances -1.IMEX methods (diagonal linear solver) +#### Implicit-Explicit Methods ```julia -abstols = 0.1 .^ (5:8) +abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = diag_linsolve), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(linsolve = diag_linsolve), :dts => 5e-3 * multipliers), - Dict(:alg => CNLF2(linsolve = diag_linsolve), :dts => 5e-3 * multipliers), - Dict(:alg => SBDF2(linsolve = diag_linsolve), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] -@time wp1 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, seconds = 5, - save_everystop = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp1, label = labels, markershape = :auto, title = "IMEX methods, diagonal linsolve, low order") +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` - 2. ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler", - "ETDRK2 (caching)") -@time wp2 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp2, label = labels, markershape = :auto, title = "ExpRK methods, low order") +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(linsolve = diag_linsolve), :dts => 5e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)] -labels = ["CNAB2 (diagonal linsolve)" "ETDRK2"] -@time wp3 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp3, label = labels, markershape = :auto, title = "Between family, low orders") +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (m=5)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances +### Low Tolerances -## In-family comparisons +#### Implicit-Explicit Methods -1.IMEX methods (band linear solver) +Dense/banded linear solvers. ```julia abstols = 0.1 .^ (7:13) reltols = 0.1 .^ (4:10) setups = [ - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1))] -labels = hcat("ARKODE3", "ARKODE4", "ARKODE5") -@time wp4 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp4, label = labels, markershape = :auto, title = "IMEX methods, band linsolve, medium order") + Dict(:alg => KenCarp3()), + Dict(:alg => KenCarp4()), + Dict(:alg => KenCarp5()), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Band, jac_upper=1, jac_lower=1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)) +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Band Linsolve, Low Tolerances") ``` -2.ExpRK methods +Krylov linear solvers. +```julia +abstols = 0.1 .^ (7:13) +reltols = 0.1 .^ (4:10) +setups = [ + Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") +``` + +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK4 (caching)", - "HochOst4 (caching)") -@time wp5 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp5, label = labels, markershape = :auto, title = "ExpRK methods, medium order") + Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers), +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) setups = [ - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)), Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers)] -labels = hcat("ARKODE (nondiagonal linsolve)", "ETDRK3 ()", "ETDRK4 ()") -#"ARKODE (Krylov linsolve)") -@time wp6 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp6, label = labels, markershape = :auto, title = "Between family, medium order") + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), +] +labels = hcat( + "ARKODE (Band linsolve)", + "ETDRK3 (caching)", + "ETDRK4 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) diff --git a/benchmarks/SimpleHandwrittenPDE/kdv_fdm_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/kdv_fdm_wpd.jmd index e8b1eb1b9..658cb95f8 100644 --- a/benchmarks/SimpleHandwrittenPDE/kdv_fdm_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/kdv_fdm_wpd.jmd @@ -1,249 +1,253 @@ --- -title: KdV FDM Work-Precision Diagrams -author: HAO HAO +title: Kortwrieg-de Vries Equation FDM Work-Precision Diagrams +author: Arjit Seth --- +### Problem Description + +The Kortwrieg de Vries (KdV) partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 20$, with the following initial and periodic boundary conditions: +```math +\begin{align} + \partial_t u(t,x) & = -6u(t,x)\partial_x u(t,x) -\partial_x^3 u(t,x), \\ + u(0,x) & = \cos\left(\frac{\pi x}{L}\right), \\ + u(t,-L) & = u(t,L) = -1. +\end{align} +``` +The spatial derivative operators are represented via finite difference approximations on an equispaced grid of points $x_s \in [-L, L]$. +```math +\begin{align} + \frac{du}{dt} & = -6uD_x u -D_x^3 u, \\ + u(0,x_s) & = \cos\left(\frac{\pi x_s}{L}\right), \\ + u(t,-L) & = u(t,L) = -1. +\end{align} +``` + +### Implementation + ```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve +using OrdinaryDiffEq using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using SummationByPartsOperators +const SBP = SummationByPartsOperators +using Plots +gr(); ``` -Here is the KdV equation using FDM. - ```julia -# Define the linear and nonlinear terms -function lin_term(N) - dx = 1/(N + 1) - du = ones(N-1) # off diagonal - du2 = -ones(N-1) # off diagonal - d = (1/2) * ones(N-2) - d2 = (-1/2) * ones(N-2) - diag=-2 * ones(N) - DiffEqArrayOperator(5e-5 * (1/dx^3) * diagm(-2 => d2, -1 => du, 1 => du2, 2 => d)) -end - -function nl_term(N) - dx = 1/(N + 1) - du = ones(N - 1) # super diagonal - dl = -ones(N - 1) # lower diagonal - D = (0.2/dx) * diagm(-1 => dl, 1 => du) - - tmp = zeros(N) - function (du, u, p, t) - @. tmp = u^2 - mul!(du, D, tmp) - end -end +nonlinear_convection!(du, u, p, t) = du .= (-6 * p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2)) # Construct the problem -function kdv(N) - f1 = lin_term(N) - f2 = nl_term(N) - dx = 1 / (N + 1) - xs = (1:N) * dx - - μ0 = 0.3; - σ0 = 0.05 - f0 = x -> 0.9*exp(-(x - μ0)^2 / (2 * σ0^2)) - u0 = f0.(xs) - prob = SplitODEProblem(f1, f2, u0, (0.0, 1.0)) - xs, prob +function kortwrieg_de_vries(N, L, alpha) + D1 = periodic_derivative_operator(derivative_order = 1, accuracy_order = 4, + xmin = -L, xmax = L, N = N) + D3 = periodic_derivative_operator(derivative_order = 3, accuracy_order = 4, + xmin = -L, xmax = L, N = N) + + # u0 = usol.(first(tspan), SBP.grid(D1)) + x = SBP.grid(D1) + u0 = @. cos(π * x / L) # Initial condition + p = (; D1, alpha) + + tspan = (0.0, 1.0) + prob = SplitODEProblem(MatrixOperator(-p.alpha * sparse(D3)), + nonlinear_convection!, + u0, tspan, p); + + return x, prob end; ``` -Reference solution using Tsit5 is below: +### Reference Solution + +Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision. ```julia -xs, prob = kdv(200) -sol = solve(prob, Tsit5(); abstol = 1e-11, reltol = 1e-11, dt = 1e-7, adptive = false) -test_sol = TestSolution(sol); - -tslices = [0.0 0.25 0.50 0.75 1.00] -ys = hcat((sol(t) for t in tslices)...) -labels = ["t = $t" for t in tslices] -fn=plot(xs, ys, label = labels) -``` +N = 128 # Number of grid points +L = 16.0 # Domain length +alpha = 5.0 # Time scaling factor +xs, prob = kortwrieg_de_vries(N, L, alpha) -Linear solvers +@time sol = solve(prob, AutoVern7(RadauIIA5(autodiff=false)); + dt = 1e-4, abstol = 1e-14, reltol = 1e-14); -```julia -const LS_Dense = LinSolveFactorize(lu) +test_sol = TestSolution(sol) # Reference solution for error estimation + +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +plt = heatmap(xs, tslices, ys', xlabel="x", ylabel="t") ``` -## High tolerances +## Work-Precision Diagrams -## In-family comparisons +### High Tolerances -1.IMEX methods (dense linear solver) +#### Implicit-Explicit Methods ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(), :dts => 1e-5 * multipliers), +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), - Dict(:alg => SBDF2(), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` -1.IMEX methods (Krylov linear solver) +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [#Dict(:alg => IMEXEuler(linsolve=KrylovJL_GMRES()), :dts => 1e-5 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-5 * multipliers), - Dict(:alg => CNLF2(linsolve = KrylovJL_GMRES()), :dts => 1e-5 * multipliers), - Dict(:alg => SBDF2(linsolve = KrylovJL_GMRES()), :dts => 1e-4 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, Krylov linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` - 2. ExpRK methods - -```julia -abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much -reltols = 0.1 .^ (1:4) -multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-6 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 5), :dts => 1e-3 * multipliers), - #Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-3 * multipliers), matrix contains Infs or NaNs - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler (caching)", "NorsettEuler (m=5)",# "NorsettEuler (m=20)", - "ETDRK2 (caching)", "ETDRK2 (m=5)", "ETDRK2 (m=20)") -@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, low order") -``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(), :dts => 1e-5 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-5 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers)] -labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"] +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, low orders") +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances - -## In-family comparisons - -1.IMEX methods (dense linear solver) - -```julia -abstols = 0.1 .^ (7:13) -reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3()), - Dict(:alg => KenCarp4()), - Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense))] -labels = hcat("KenCarp3", "KenCarp4", "KenCarp5", "ARKODE3", "ARKODE4", "ARKODE5") -@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, medium order") -``` +### Low Tolerances -1.IMEX methods (Krylov linear solver) +#### Implicit-Explicit Methods +Krylov linear solvers. ```julia -abstols = 0.1 .^ (7:13) -reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES))] -labels = ["KenCarp3" "KenCarp4" "KenCarp5" "ARKODE3" "ARKODE4" "ARKODE5"] +abstols = 0.1 .^ (7:11) +reltols = 0.1 .^ (4:8) +setups = [ + # KenCarp methods take forever with adaptive timestepping for some reason + # Dict(:alg => KenCarp3()), + # Dict(:alg => KenCarp4()), + # Dict(:alg => KenCarp5()), + # Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + # Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + # Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + # "KenCarp3", + # "KenCarp4", + # "KenCarp5", + # "KenCarp3 (Krylov)", + # "KenCarp4 (Krylov)", + # "KenCarp5 (Krylov)", + "ARKODE3 (Krylov)", + "ARKODE4 (Krylov)", + "ARKODE5 (Krylov)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, medium order") +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - #Dict(:alg => ETDRK3(krylov=true, m=5), :dts => 1e-2 * multipliers),matrix contains Infs or NaNs - Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers), - #Dict(:alg => ETDRK4(krylov=true, m=5), :dts => 1e-2 * multipliers),matrix contains Infs or NaNs - Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers)] -#Dict(:alg => HochOst4(krylov=true, m=5), :dts => 1e-2 * multipliers)] matrix contains Infs or NaNs -labels = hcat("ETDRK3 (caching)", "ETDRK4 (caching)",# "ETDRK3 (m=5)", "ETDRK4 (m=5)" - "HochOst4 (caching)")#, "HochOst4 (m=5)") +setups = [Dict(:alg => ETDRK3(), :dts => 1e-3 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers), + Dict(:alg => HochOst4(), :dts => 1e-3 * multipliers)] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, medium order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense)), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES)), +setups = [ + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)), Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers)] + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), +] labels = hcat( - "KenCarp5 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp5 (Krylov linsolve)", - "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)") + "ARKODE (Band linsolve)", + "ETDRK3 (caching)", + "ETDRK4 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, medium order") +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) diff --git a/benchmarks/SimpleHandwrittenPDE/kdv_spectral_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/kdv_spectral_wpd.jmd index 3746535f3..af7cb1268 100644 --- a/benchmarks/SimpleHandwrittenPDE/kdv_spectral_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/kdv_spectral_wpd.jmd @@ -1,179 +1,249 @@ --- -title: KdV Pseudospectral Methods Work-Precision Diagrams -author: HAO HAO +title: Korteweg-De Vries Equation Pseudospectral Methods Work-Precision Diagrams +author: Arjit Seth --- ```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve +using OrdinaryDiffEq using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using SummationByPartsOperators +const SBP = SummationByPartsOperators +using Plots gr() ``` -Here is the Burgers equation using Fourier spectral methods. +## Problem Description + +The Kortwrieg de Vries (KdV) partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 5$, with the following initial and periodic boundary conditions: +```math +\begin{align} + \partial_t u(t,x) & = -6u(t,x)\partial_x u(t,x) -\partial_x^3 u(t,x), \\ + u(0,x) & = \cos\left(\frac{\pi x}{L}\right), \\ + u(t,-L) & = u(t,L) = -1. +\end{align} +``` +The spatial derivative operators are represented via Fourier pseudospectral approximations. Here, the domain is discretized by projecting on an equispaced grid of points $x_s \in [-L, L]$; the solution is approximated on this grid via linear combinations of sinusoidal functions in space. +```math +\begin{align} + \frac{du}{dt} & = -6uD_x u -D_x^3 u, \\ + u(0,x_s) & = \cos\left(\frac{\pi x_s}{L}\right), \\ + u(t,-L) & = u(t,L) = -1. +\end{align} +``` + + +### Implementation ```julia -S = Fourier() -n = 512 -x = points(S, n) -D2 = Derivative(S, 2)[1:n, 1:n] -D = (Derivative(S) → S)[1:n, 1:n] -T = ApproxFun.plan_transform(S, n) -Ti = ApproxFun.plan_itransform(S, n) - -D3 = (Derivative(S, 3) → S)[1:n, 1:n] -û₀ = T*cos.(x) -δ=0.022 -tmp = similar(û₀) -p = (D, T, Ti, similar(tmp), tmp) -function kdv(dû, û, p, t) - D, T, Ti, u, tmp = p - mul!(u, D, û) - mul!(tmp, Ti, u) - mul!(u, Ti, û) - @. tmp=u*tmp - mul!(u, T, tmp) - @.dû = 6*u -end +nonlinear_convection!(du, u, p, t) = du .= (-6 * p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2)) + +function korteweg_de_vries(N, L, alpha) + D1 = fourier_derivative_operator(xmin = -L, xmax = L, N = N) + D3 = D1^3 # 3rd derivative operator + x = SBP.grid(D1) + u0 = @. cos(π * x / L) # Initial condition + p = (; D1, alpha) + + tspan = (0.0, 1.0) + prob = SplitODEProblem(MatrixOperator(-p.alpha * Matrix(D3)), + nonlinear_convection!, + u0, tspan, p); + + return x, prob +end; ``` -Reference solution using RadauIIA5 is below: +### Reference Solution + +Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision. ```julia -prob = SplitODEProblem(DiffEqArrayOperator(-D3), kdv, û₀, (0.0, 5.0), p) -sol = solve(prob, RadauIIA5(autodiff = false); reltol = 1e-14, abstol = 1e-14) -test_sol = TestSolution(sol) - -tslices=[0.0 1.0 2.0 3.0 5.0] -ys=hcat((Ti*sol(t) for t in tslices)...) -labels=["t=$t" for t in tslices] -plot(x, ys, label = labels) +L = 16.0 # Domain length +n = 128 # Number of Chebyshev points +alpha = 5.0 # Time scaling factor +xs, prob = korteweg_de_vries(n, L, alpha) + +@time sol = solve(prob, AutoVern7(RadauIIA5(autodiff=false)); + dt = 1e-4, reltol = 1e-12, abstol = 1e-12); + +test_sol = TestSolution(sol) # Reference solution for error estimation + +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +plt = heatmap(xs, tslices, ys', xlabel = "x", ylabel = "t") ``` -## High tolerances +## Work-Precision Diagrams -## In-family comparisons +### High Tolerances -1.IMEX methods (diagonal linear solver) +#### Implicit-Explicit Methods ```julia -abstols = 0.1 .^ (5:8) +abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = diag_linsolve), :dts => 1e-5 * multipliers), - Dict(:alg => CNAB2(linsolve = diag_linsolve), :dts => 5e-7 * multipliers), - Dict(:alg => CNLF2(linsolve = diag_linsolve), :dts => 5e-7 * multipliers), - Dict(:alg => SBDF2(linsolve = diag_linsolve), :dts => 1e-5 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] -@time wp1 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, seconds = 5, - save_everystop = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp1, label = labels, markershape = :auto, title = "IMEX methods, diagonal linsolve, low order") +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` - 2. ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler", - "ETDRK2 (caching)") -@time wp2 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp2, label = labels, markershape = :auto, title = "ExpRK methods, low order") +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` -## Between family comparisons + +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(linsolve = diag_linsolve), :dts => 5e-5 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)] -labels = ["CNAB2 (diagonal linsolve)" "ETDRK2"] -@time wp3 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp3, label = labels, markershape = :auto, title = "Between family, low orders") +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances +### Low Tolerances -## In-family comparisons +#### Implicit-Explicit Methods -1.IMEX methods (band linear solver) ```julia -abstols = 0.1 .^ (7:13) -reltols = 0.1 .^ (4:10) +abstols = 0.1 .^ (8:10) +reltols = 0.1 .^ (5:7) setups = [ - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1))] -labels = hcat("ARKODE3", "ARKODE4", "ARKODE5") -@time wp4 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp4, label = labels, markershape = :auto, title = "IMEX methods, band linsolve, medium order") + Dict(:alg => KenCarp3()), + Dict(:alg => KenCarp4()), + # Dict(:alg => KenCarp5()), + Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + # Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + # "KenCarp5", + "KenCarp3 (Krylov)", + "KenCarp4 (Krylov)", + # "KenCarp5 (Krylov)", + "ARKODE3 (Krylov)", + "ARKODE4 (Krylov)", + "ARKODE5 (Krylov)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK4 (caching)", - "HochOst4 (caching)")#,"ETDRK4 (m=5)" "ETDRK3 (m=5)" "HochOst4 (m=5)") -@time wp5 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp5, label = labels, markershape = :auto, title = "ExpRK methods, medium order") +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-3 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers), + Dict(:alg => HochOst4(), :dts => 1e-3 * multipliers), +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) setups = [ - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers)] -labels = hcat("ARKODE (nondiagonal linsolve)", "ETDRK3 ()", "ETDRK4 ()") -@time wp6 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp6, label = labels, markershape = :auto, title = "Between family, medium order") + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), + Dict(:alg => ETDRK3(), :dts => 1e-3 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers), +] +labels = hcat( + "ARKODE (Krylov linsolve)", + "ETDRK3 (caching)", + "ETDRK4 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` ```julia, echo = false using SciMLBenchmarks -SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) +SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder],WEAVE_ARGS[:file]) ``` diff --git a/benchmarks/SimpleHandwrittenPDE/ks_fdm_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/ks_fdm_wpd.jmd index 008c83821..251d1b522 100644 --- a/benchmarks/SimpleHandwrittenPDE/ks_fdm_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/ks_fdm_wpd.jmd @@ -1,254 +1,244 @@ --- -title: KS FDM Work-Precision Diagrams -author: HAO HAO +title: Kuramoto-Sivashinsky Finite-Difference Method Work-Precision Diagrams +author: Arjit Seth --- +## Problem Description + +The Kuramoto-Sivashinsky partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 30$, with the following initial and periodic boundary conditions: +```math +\begin{align} + \partial_t u & = -\kappa u(t,x)\partial_x u(t,x) - \beta \partial_x^2 u(t,x) - \gamma \partial_x^4 u(t,x), \\ + u(0,x) & = \cos\left(\frac{2\pi}{L} x\right), \\ + u(t,-L) & = u(t,L) = 1. +\end{align} +``` +The spatial derivative operators are represented via finite difference approximations, incorporating the periodic boundary conditions on an equispaced grid of points $x_s \in [-L, L]$. The coefficients $\kappa = 1,~\beta = 1/2,~\gamma = 1/8$ are chosen to produce `interesting' behavior as seen in the reference solution below. +```math +\begin{align} + \frac{du}{dt} & = -\kappa u D_x u - \beta D_x^2 u - \gamma D_x^4 u, \\ + u(0,x_s) & = \cos\left(\frac{2\pi}{L} x_s\right), \\ + u(t,-L) & = u(t,L) = 1. +\end{align} +``` + +### Implementation + ```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve +using OrdinaryDiffEq using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using SummationByPartsOperators +const SBP = SummationByPartsOperators +using Plots +gr(); ``` -Here is the KS equation using FDM. - ```julia -# Define the linear and nonlinear terms -function lin_term(N) - #which is -(D2+D4) - dx = 1/(N + 1) - d2 = (-2) * ones(N) # main diagonal - du2 = ones(N - 1) # off diagonal - - d4 = 6 * ones(N) # main diagonal - du4 = (-4) * ones(N - 1) # off diagonal - duu4 = ones(N - 2) - DiffEqArrayOperator(-0.0004*((1/dx^2) * diagm(-1 => du2, 0 => d2, 1 => du2) - + - (1/dx^4) * diagm( - -2 => duu4, -1 => du4, 0 => d4, 1 => du4, 2 => duu4))) -end - -function nl_term(N) - dx = 1/(N + 1) - du = ones(N - 1) # super diagonal - dl = -ones(N - 1) # lower diagonal - D = (-0.2/(4*dx)) * diagm(-1 => dl, 1 => du) - - tmp = zeros(N) - function (du, u, p, t) - @. tmp = u^2 - mul!(du, D, tmp) - end -end +nonlinear_convection!(du, u, p, t) = du .= (-p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2)) # Construct the problem -function ks(N) - f1 = lin_term(N) - f2 = nl_term(N) - dx = 1 / (N + 1) - xs = (1:N) * dx - - μ0 = 0.3; - σ0 = 0.05 - f0 = x -> 0.6*exp(-(x - μ0)^2 / (2 * σ0^2)) - u0 = f0.(xs) - prob = SplitODEProblem(f1, f2, u0, (0.0, 1.0)) - xs, prob +function kuramoto_sivashinsky(N, L, alpha) + D1 = periodic_derivative_operator(derivative_order = 1, accuracy_order = 4, + xmin = -L, xmax = L, N = N) + D2 = periodic_derivative_operator(derivative_order = 2, accuracy_order = 4, + xmin = -L, xmax = L, N = N) + D4 = periodic_derivative_operator(derivative_order = 4, accuracy_order = 4, + xmin = -L, xmax = L, N = N) + + x = SBP.grid(D1) + u0 = @. cos(2π * x / L) # Initial condition + p = (; D1, alpha) + + tspan = (0.0, 1.0) + prob = SplitODEProblem(MatrixOperator(-p.alpha / 2 * (sparse(D2) + 1/4 * sparse(D4))), + nonlinear_convection!, + u0, tspan, p); + + return x, prob end; ``` -Reference solution using RadauIIA5 is below: +### Reference Solution -```julia -xs, prob = ks(200) -sol = solve(prob, RadauIIA5(autodiff = false); abstol = 1e-14, reltol = 1e-14) -test_sol = TestSolution(sol); +Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision. -tslices = [0.0 0.25 0.50 0.75 1.0] -ys = hcat((sol(t) for t in tslices)...) -labels = ["t = $t" for t in tslices] -plot(xs, ys, label = labels) -``` +```julia +N = 128 # Number of grid points +L = 16.0 # Domain length +alpha = 30.0 # Time scaling factor +xs, prob = kuramoto_sivashinsky(N, L, alpha) +@time sol = solve(prob, RadauIIA5(autodiff=false); dt = 1e-4, abstol=1e-14, reltol=1e-14, adaptive=true) -Linear solvers +test_sol = TestSolution(sol); -```julia -const LS_Dense = LinSolveFactorize(lu) +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +plt = heatmap(xs, tslices, ys', xlabel="x", ylabel="t") ``` -## High tolerances +## Work-Precision Diagrams -## In-family comparisons +### High Tolerances -1.IMEX methods (dense linear solver) +#### Implicit-Explicit Methods ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(), :dts => 1e-3 * multipliers), +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), - Dict(:alg => SBDF2(), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` -1.IMEX methods (Krylov linear solver) +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = KrylovJL_GMRES()), :dts => 1e-9 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-9 * multipliers), - Dict(:alg => CNLF2(linsolve = KrylovJL_GMRES()), :dts => 1e-9 * multipliers), - Dict(:alg => SBDF2(linsolve = KrylovJL_GMRES()), :dts => 1e-9 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), # Maxiters + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), # Maxiters + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, Krylov linsolve, low order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` - 2. ExpRK methods -```julia -abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much -reltols = 0.1 .^ (1:4) -multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 5), :dts => 1e-3 * multipliers), - Dict(:alg => NorsettEuler(krylov = true, m = 20), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK2(krylov = true, m = 20), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler (caching)", "NorsettEuler (m=5)", "NorsettEuler (m=20)", - "ETDRK2 (caching)", "ETDRK2 (m=5)", "ETDRK2 (m=20)") -@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, low order") -``` - -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), - Dict(:alg => CNAB2(linsolve = KrylovJL_GMRES()), :dts => 1e-9 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers)] -labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"] +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, low orders") +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances - -## In-family comparisons - -1.IMEX methods (dense linear solver) - -```julia -abstols = 0.1 .^ (7:13) -reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3()), - Dict(:alg => KenCarp4()), - Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Dense)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense))] -labels = hcat("KenCarp3", "KenCarp4", "KenCarp5", "ARKODE3", "ARKODE4", "ARKODE5") -@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); +### Low Tolerances -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, dense linsolve, medium order") -``` +#### Implicit-Explicit Methods -1.IMEX methods (Krylov linear solver) +Krylov linear solvers. ```julia abstols = 0.1 .^ (7:13) reltols = 0.1 .^ (4:10) -setups = [Dict(:alg => KenCarp3(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :GMRES)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES))] -labels = ["KenCarp3" "KenCarp4" "KenCarp5" "ARKODE3" "ARKODE4" "ARKODE5"] +setups = [ + # KenCarp methods take forever with adaptive timestepping for some reason + # Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + # Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + # Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + # "KenCarp3", + # "KenCarp4", + # "KenCarp5", + "ARKODE3", + "ARKODE4", + "ARKODE5", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "IMEX methods, medium order") +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers), Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK3 (m=5)", "ETDRK4 (caching)", - "ETDRK4 (m=5)", "HochOst4 (caching)", "HochOst4 (m=5)") +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "ExpRK methods, medium order") +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => KenCarp5()), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Dense)), - Dict(:alg => KenCarp5(linsolve = KrylovJL_GMRES())), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :GMRES)), - Dict(:alg => ETDRK3(krylov = true, m = 5), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(krylov = true, m = 5), :dts => 1e-2 * multipliers)] -labels = hcat( - "KenCarp5 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp5 (Krylov linsolve)", - "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)") +setups = [ + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), +] +labels = hcat("ARKODE (Krylov linsolve)", "ETDRK3 ()", "ETDRK4 ()") @time wp = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); -plot(wp, label = labels, markershape = :auto, title = "Between family, medium order") +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file]) diff --git a/benchmarks/SimpleHandwrittenPDE/ks_spectral_wpd.jmd b/benchmarks/SimpleHandwrittenPDE/ks_spectral_wpd.jmd index 5c5112296..1850fbb93 100644 --- a/benchmarks/SimpleHandwrittenPDE/ks_spectral_wpd.jmd +++ b/benchmarks/SimpleHandwrittenPDE/ks_spectral_wpd.jmd @@ -1,188 +1,245 @@ --- -title: KS Pseudospectral Methods Work-Precision Diagrams -author: HAO HAO +title: Kuramoto-Sivashinsky Pseudospectral Methods Work-Precision Diagrams +author: Arjit Seth --- -```julia -using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve -using DiffEqDevTools -using LinearAlgebra, RecursiveFactorization -using Plots; -gr() +## Problem Description + +The Kuramoto-Sivashinsky partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 10$, with the following initial and periodic boundary conditions: +```math +\begin{align} + \partial_t u & = -\kappa u(t,x) \partial_x u(t,x) - \beta \partial_x^2 u(t,x) - \gamma \partial_x^4 u(t,x) \\ + u(0,x) & = \cos\left(\frac{2\pi}{L} x\right), \\ + u(t,-L) & = u(t,L) = 1. +\end{align} +``` +The spatial derivative operators are represented via Fourier pseudospectral approximations. Here, the domain is discretized by projecting on an equispaced grid of points $x_s \in [-L, L]$; the solution is approximated on this grid via linear combinations of sinusoidal functions in space. The coefficients $\kappa = 1, \beta = 1/2,~\gamma = 1/16$ are chosen to produce `interesting' behavior as seen in the reference solution below. +```math +\begin{align} + \frac{du}{dt} & = -\kappa u D_x u - \beta D_x^2 u - \gamma D_x^4 u \\ + u(0,x_s) & = \cos\left(\frac{2\pi}{L} x_s\right), \\ + u(t,-L) & = u(t,L) = 1. +\end{align} ``` -Here is the kuramoto_sivashinsky equation using Fourier spectral methods. +### Implementation ```julia -S = Fourier() -n = 512 -x = points(S, n) -D2 = Derivative(S, 2)[1:n, 1:n] -D = (Derivative(S) → S)[1:n, 1:n] -T = ApproxFun.plan_transform(S, n) -Ti = ApproxFun.plan_itransform(S, n) - -D4 = Derivative(S, 4)[1:n, 1:n] -û₀ = T*(cos.(x ./ 16) .* (1 .+ sin.(x ./ 2.04))) -tmp=similar(û₀) -q = (D, T, Ti, tmp, similar(tmp), similar(tmp)) -function kuramoto_sivashinsky(dû, û, q, t) - D, T, Ti, tmp, u, uc = q - mul!(u, D, û) - mul!(tmp, Ti, u) - mul!(u, Ti, û) - @. tmp=tmp*u - mul!(u, T, tmp) - #mul!(uc, D2, û) - @. dû = - u -end +using OrdinaryDiffEq +using DiffEqDevTools +using SciMLOperators +using LinearSolve +using LinearAlgebra +using SparseArrays +using Sundials +using SummationByPartsOperators +const SBP = SummationByPartsOperators +using Plots +gr(); ``` -Reference solution using Rodas5 is below: - ```julia -prob = SplitODEProblem( - DiffEqArrayOperator(-Diagonal(D4+D2)), kuramoto_sivashinsky, û₀, (0.0, 5.0), q) -sol = solve(prob, RadauIIA5(autodiff = false); reltol = 1e-14, abstol = 1e-14) -test_sol = TestSolution(sol) - -tslices=[0.0 1.0 2.0 3.0 5.0] -ys=hcat((Ti*sol(t) for t in tslices)...) -labels=["t=$t" for t in tslices] -plot(x, ys, label = labels) +nonlinear_convection!(du, u, p, t) = du .= (-p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2)) + +function kuramoto_sivashinsky(N, L, alpha) + D1 = fourier_derivative_operator(xmin = -L, xmax = L, N = N) + D2 = D1^2 # 2nd derivative operator + D4 = D1^4 # 4th derivative operator + x = SBP.grid(D1) + u0 = @. cos(2π * x / L) # Initial condition + p = (; D1, alpha) + + tspan = (0.0, 1.0) + prob = SplitODEProblem(MatrixOperator(-p.alpha / 2 * (Matrix(D2) + 1/8 * Matrix(D4))), + nonlinear_convection!, + u0, tspan, p); + + return x, prob +end; ``` -## High tolerances +### Reference Solution + +Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision. ```julia -diag_linsolve=LinSolveFactorize(W->let tmp = tmp - for i in 1:size(W, 1) - tmp[i] = W[i, i] - end - Diagonal(tmp) -end) +N = 128 # Number of grid points +L = 16.0 # Domain length +alpha = 10.0 # Time scaling factor +xs, prob = kuramoto_sivashinsky(N, L, alpha) + +sol = solve(prob, Rodas5(autodiff=false); dt = 1e-4, reltol = 1e-12, abstol = 1e-12) + +test_sol = TestSolution(sol) # Reference solution for error estimation + +tslices = LinRange(prob.tspan..., 50) +ys = mapreduce(sol, hcat, tslices) +heatmap(xs, tslices, ys', xlabel = "x", ylabel = "t") ``` -## In-family comparisons +## Work-Precision Diagrams + +### High Tolerances -1.IMEX methods (diagonal linear solver) +#### Implicit-Explicit Methods ```julia -abstols = 0.1 .^ (5:8) +abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => IMEXEuler(linsolve = diag_linsolve), :dts => 1e-3 * multipliers), - Dict(:alg => CNAB2(linsolve = diag_linsolve), :dts => 5e-3 * multipliers), - Dict(:alg => CNLF2(linsolve = diag_linsolve), :dts => 5e-3 * multipliers), - Dict(:alg => SBDF2(linsolve = diag_linsolve), :dts => 1e-3 * multipliers)] -labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"] -@time wp1 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, seconds = 5, - save_everystop = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp1, label = labels, markershape = :auto, title = "IMEX methods, diagonal linsolve, low order") +setups = [ + Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers), + Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "IMEXEuler", + "CNAB2", + "CNLF2", + "SBDF2", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance") ``` - 2. ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => NorsettEuler(), :dts => 1e-3 * multipliers), - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)] -labels = hcat("NorsettEuler", - "ETDRK2 (caching)") -@time wp2 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp2, label = labels, markershape = :auto, title = "ExpRK methods, low order") +setups = [ + Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers) +] +labels = hcat( + "NorsettEuler (caching)", + "NorsettEuler (m=5)", + "NorsettEuler (m=20)", + "ETDRK2 (caching)", + "ETDRK2 (m=5)", + "ETDRK2 (m=20)" +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance") ``` -## Between family comparisons + +#### Comparisons Between Families ```julia abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (1:4) multipliers = 0.5 .^ (0:3) -setups = [Dict(:alg => CNAB2(linsolve = diag_linsolve), :dts => 5e-3 * multipliers) - Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)] -labels = ["CNAB2 (diagonal linsolve)" "ETDRK2"] -@time wp3 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp3, label = labels, markershape = :auto, title = "Between family, low orders") +setups = [ + Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers), + Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers), + Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers), +] +labels = hcat( + "CNAB2 (dense linsolve)", + "CNAB2 (Krylov linsolve)", + "ETDRK2 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances") ``` -## Low tolerances -## In-family comparisons +### Low Tolerances -1.IMEX methods (band linear solver) +#### Implicit-Explicit Methods ```julia -abstols = 0.1 .^ (7:13) -reltols = 0.1 .^ (4:10) +abstols = 0.1 .^ (7:10) +reltols = 0.1 .^ (4:7) setups = [ - Dict(:alg => ARKODE(Sundials.Implicit(), order = 3, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 4, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1))] -labels = hcat("ARKODE3", "ARKODE4", "ARKODE5") -@time wp4 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp4, label = labels, markershape = :auto, title = "IMEX methods, band linsolve, medium order") + Dict(:alg => KenCarp3()), + Dict(:alg => KenCarp4()), + Dict(:alg => KenCarp5()), + Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())), + Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())), + # ARKODE with banded linear solvers fail. + Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), +] +labels = hcat( + "KenCarp3", + "KenCarp4", + "KenCarp5", + "KenCarp3 (Krylov)", + "KenCarp4 (Krylov)", + "KenCarp5 (Krylov)", + "ARKODE3 (Krylov)", + "ARKODE4 (Krylov)", + "ARKODE5 (Krylov)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances") ``` -2.ExpRK methods +#### Exponential Integrators ```julia abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) -setups = [Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), +setups = [ + Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), - Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers)] -labels = hcat("ETDRK3 (caching)", "ETDRK4 (caching)", - "HochOst4 (caching)") -@time wp5 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp5, label = labels, markershape = :auto, title = "ExpRK methods, medium order") + Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers), +] +labels = hcat( + "ETDRK3 (caching)", + "ETDRK4 (caching)", + "HochOst4 (caching)", +) +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances") ``` -## Between family comparisons +#### Comparisons Between Families ```julia abstols = 0.1 .^ (7:11) reltols = 0.1 .^ (4:8) multipliers = 0.5 .^ (0:4) setups = [ - Dict(:alg => ARKODE(Sundials.Implicit(), order = 5, linear_solver = :Band, - jac_upper = 1, jac_lower = 1)), + Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)), Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers), - Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers)] -labels = hcat("ARKODE (nondiagonal linsolve)", "ETDRK3 ()", "ETDRK4 ()") -@time wp6 = WorkPrecisionSet(prob, abstols, reltols, setups; - print_names = true, names = labels, - numruns = 5, error_estimate = :l2, - save_everystep = false, appxsol = test_sol, maxiters = Int(1e5)); - -plot(wp6, label = labels, markershape = :auto, title = "Between family, medium order") + Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers), +] +labels = hcat("ARKODE (Krylov linsolve)", "ETDRK3 ()", "ETDRK4 ()") +@time wp = WorkPrecisionSet(prob, abstols, reltols, setups; + print_names=true, names=labels, numruns=5, error_estimate=:l2, + save_everystep=false, appxsol=test_sol, maxiters=Int(1e5)); + +plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances") ``` + ```julia, echo = false using SciMLBenchmarks SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file])