Move SDE default algorithm from DifferentialEquations.jl to StochasticDiffEq.jl

This PR moves the default SDE solver implementation from DifferentialEquations.jl to StochasticDiffEq.jl, following the pattern established in SciML/DelayDiffEq.jl#326 and SciML/DelayDiffEq.jl#334.

## Changes
- Added `src/default_sde_alg.jl` containing the default algorithm selection logic
- Implemented `__init` and `__solve` dispatches for `SDEProblem` with `Nothing` algorithm
- Added `get_alg_hints` helper function for extracting algorithm hints from kwargs
- Added comprehensive tests in `test/default_solver_test.jl`
- Updated module to include the new default algorithm file

## Default Algorithm Behavior
When no algorithm is specified, the solver now automatically selects:
- SOSRI() as the standard default
- RKMilCommute() for commutative noise
- ImplicitRKMil() for stiff problems or non-identity mass matrices
- RKMil() for Stratonovich interpretation
- LambaEM() / LambaEulerHeun() for non-diagonal noise
- ISSEM() / ImplicitEulerHeun() for stiff non-diagonal problems
- SOSRA() / SKenCarp() for additive noise

## Test Plan
- [x] Added tests verifying default solver dispatch
- [x] Tests verify correct algorithm selection for various problem types
- [x] All tests pass locally

This is part of the ongoing effort to modularize DifferentialEquations.jl by moving default solvers to their respective packages.

🤖 Generated with [Claude Code](https://claude.ai/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

Author: ChrisRackauckas

src/StochasticDiffEq.jl

@@ -158,6 +158,7 @@ include("iterated_integrals.jl")
 include("SROCK_utils.jl")
 include("composite_algs.jl")
 include("weak_utils.jl")
+include("default_sde_alg.jl")
 
 export StochasticDiffEqAlgorithm, StochasticDiffEqAdaptiveAlgorithm,
        StochasticCompositeAlgorithm

src/default_sde_alg.jl

@@ -0,0 +1,83 @@
+# Default algorithm selection for SDEs
+# Moved from DifferentialEquations.jl as part of modularization effort
+
+using LinearAlgebra: I
+
+# Helper function to extract alg_hints from keyword arguments
+function get_alg_hints(o)
+    :alg_hints ∈ keys(o) ? alg_hints = o[:alg_hints] : alg_hints = Symbol[:auto]
+end
+
+function default_algorithm(
+        prob::DiffEqBase.AbstractSDEProblem{uType, tType, isinplace, ND};
+        kwargs...) where {uType, tType, isinplace, ND}
+    o = Dict{Symbol, Any}(kwargs)
+    extra_kwargs = Any[]
+    alg = SOSRI() # Standard default
+    uEltype = eltype(prob.u0)
+
+    alg_hints = get_alg_hints(o)
+
+    if :commutative ∈ alg_hints
+        alg = RKMilCommute()
+    end
+
+    is_stiff = :stiff ∈ alg_hints
+    is_stratonovich = :stratonovich ∈ alg_hints
+    if is_stiff || prob.f.mass_matrix !== I
+        alg = ImplicitRKMil(autodiff = false)
+    end
+
+    if is_stratonovich
+        if is_stiff || prob.f.mass_matrix !== I
+            alg = ImplicitRKMil(autodiff = false,
+                interpretation = SciMLBase.AlgorithmInterpretation.Stratonovich)
+        else
+            alg = RKMil(interpretation = SciMLBase.AlgorithmInterpretation.Stratonovich)
+        end
+    end
+
+    if prob.noise_rate_prototype != nothing || prob.noise != nothing
+        if is_stratonovich
+            if is_stiff || prob.f.mass_matrix !== I
+                alg = ImplicitEulerHeun(autodiff = false)
+            else
+                alg = LambaEulerHeun()
+            end
+        else
+            if is_stiff || prob.f.mass_matrix !== I
+                alg = ISSEM(autodiff = false)
+            else
+                alg = LambaEM()
+            end
+        end
+    end
+
+    if :additive ∈ alg_hints
+        if is_stiff || prob.f.mass_matrix !== I
+            alg = SKenCarp(autodiff = false)
+        else
+            alg = SOSRA()
+        end
+    end
+
+    # If adaptivity is not set and the tType is not a float, turn off adaptivity
+    # Bad interaction with ForwardDiff
+    #!(tType <: AbstractFloat) && (:adaptive ∉ keys(o)) && push!(extra_kwargs,:adaptive=>false)
+
+    alg, extra_kwargs
+end
+
+# Dispatch for __init with Nothing algorithm - use default
+function DiffEqBase.__init(
+        prob::DiffEqBase.AbstractSDEProblem, ::Nothing, args...; kwargs...)
+    alg, extra_kwargs = default_algorithm(prob; kwargs...)
+    DiffEqBase.__init(prob, alg, args...; extra_kwargs..., kwargs...)
+end
+
+# Dispatch for __solve with Nothing algorithm - use default
+function DiffEqBase.__solve(
+        prob::DiffEqBase.AbstractSDEProblem, ::Nothing, args...; kwargs...)
+    alg, extra_kwargs = default_algorithm(prob; kwargs...)
+    DiffEqBase.__solve(prob, alg, args...; extra_kwargs..., kwargs...)
+end

test/default_solver_test.jl

@@ -0,0 +1,58 @@
+using StochasticDiffEq, Test
+import SciMLBase
+using Random
+
+# Additive SDE test problem
+f_additive(u, p, t) = @. p[2] / sqrt(1 + t) - u / (2 * (1 + t))
+σ_additive(u, p, t) = @. p[1] * p[2] / sqrt(1 + t)
+p = (0.1, 0.05)
+additive_analytic(u0, p, t, W) = @. u0 / sqrt(1 + t) + p[2] * (t + p[1] * W) / sqrt(1 + t)
+ff_additive = SDEFunction(f_additive, σ_additive, analytic = additive_analytic)
+prob_sde_additive = SDEProblem(ff_additive, σ_additive, 1.0, (0.0, 1.0), p)
+
+Random.seed!(100)
+
+# Test default (no algorithm specified) - should use SOSRI
+prob = prob_sde_additive
+sol = solve(prob, dt = 1 / 2^(3))
+@test sol.alg isa SOSRI
+
+# Test with :additive hint - should use SOSRA
+sol = solve(prob, dt = 1 / 2^(3), alg_hints = [:additive])
+@test sol.alg isa SOSRA
+
+# Test with :stratonovich hint - should use RKMil with Stratonovich interpretation
+sol = solve(prob, dt = 1 / 2^(3), alg_hints = [:stratonovich])
+@test SciMLBase.alg_interpretation(sol.alg) ==
+      SciMLBase.AlgorithmInterpretation.Stratonovich
+@test sol.alg isa RKMil
+
+# Non-diagonal noise test problem
+f = (du, u, p, t) -> du .= 1.01u
+g = function (du, u, p, t)
+    du[1, 1] = 0.3u[1]
+    du[1, 2] = 0.6u[1]
+    du[1, 3] = 0.9u[1]
+    du[1, 4] = 0.12u[2]
+    du[2, 1] = 1.2u[1]
+    du[2, 2] = 0.2u[2]
+    du[2, 3] = 0.3u[2]
+    du[2, 4] = 1.8u[2]
+end
+prob = SDEProblem(f, g, ones(2), (0.0, 1.0), noise_rate_prototype = zeros(2, 4))
+
+# Test default with non-diagonal noise - should use LambaEM
+sol = solve(prob, dt = 1 / 2^(3))
+@test sol.alg isa LambaEM
+
+# Test with :stiff hint - should use ISSEM
+sol = solve(prob, dt = 1 / 2^(3), alg_hints = [:stiff])
+@test sol.alg isa ISSEM
+
+# Test with :additive hint - should still use SOSRA (overrides non-diagonal)
+sol = solve(prob, dt = 1 / 2^(3), alg_hints = [:additive])
+@test sol.alg isa SOSRA
+
+# Test with :stratonovich hint - should use LambaEulerHeun
+sol = solve(prob, dt = 1 / 2^(3), alg_hints = [:stratonovich])
+@test sol.alg isa LambaEulerHeun

test/runtests.jl

@@ -15,6 +15,9 @@ const is_APPVEYOR = Sys.iswindows() && haskey(ENV, "APPVEYOR")
 
 @time begin
     if GROUP == "All" || GROUP == "Interface1"
+        @time @safetestset "Default Solver Tests" begin
+            include("default_solver_test.jl")
+        end
         @time @safetestset "First Rand Tests" begin
             include("first_rand_test.jl")
         end