add quick convergence analysis for TRBDF2 as DIRK

Author: MarcoArtiano

examples/trixi_convergence.jl

@@ -0,0 +1,56 @@
+# # Using the an implicit solver based on Ariadne with Trixi.jl
+
+using Trixi
+using Implicit
+using CairoMakie
+
+
+# Notes:
+# Must disable both Polyester and LoopVectorization for Enzyme to be able to differentiate Trixi.jl
+# Using https://github.com/trixi-framework/Trixi.jl/pull/2295
+#
+# LocalPreferences.jl
+# ```toml
+# [Trixi]
+# loop_vectorization = false
+# polyester = false
+# ```
+
+@assert !Trixi._PREFERENCE_POLYESTER
+@assert !Trixi._PREFERENCE_LOOPVECTORIZATION
+
+trixi_include(joinpath(examples_dir(), "tree_2d_dgsem", "elixir_advection_basic.jl"), cfl = 1.25/2, sol = nothing, save_solution = nothing, 
+                                                                 initial_refinement_level = 6, advection_velocity = (1.0, 1.0));
+
+ode = semidiscretize(semi, (0.0, 2.0))
+
+###############################################################################
+# run the simulation
+
+sol = solve(
+	ode, 
+    # Implicit.RKImplicitEuler();
+    # Implicit.KS22();
+    # Implicit.C23();
+    # Implicit.L33();
+      Implicit.RKTRBDF2();
+    dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+	ode_default_options()..., callback = callbacks,
+	# verbose=1,
+	krylov_algo = :gmres,
+);
+
+cfl_vec = [10.0, 5.0, 2.5, 1.25, 1.25/2]
+l2_vec = [1.07287321e-02, 2.69376612e-03, 6.75387935e-04, 1.68974528e-04, 4.22696680e-05]
+linf_vec = [1.51727157e-02, 3.80957386e-03, 9.55254494e-04, 2.38966657e-04, 5.97799432e-05]
+f = Figure()
+ax = Axis(f[1, 1], xlabel = "CFL", ylabel = "Error", 
+    xscale = log10, yscale = log10, title = "Convergence Plot")
+
+lines!(ax, cfl_vec, l2_vec, label = "L2 norm")
+lines!(ax, cfl_vec, linf_vec, label = "L∞ norm")
+lines!(ax, cfl_vec, 1e-4.*cfl_vec.^2, label = "2nd order")
+
+axislegend(ax; position = :rb)
+
+f