Add default SDE algorithm with type-stable approach

[ChrisRackauckas]

Summary

This PR adds a default solver algorithm for StochasticDiffEq.jl similar to OrdinaryDiffEq.jl's OrdinaryDiffEqDefault's DefaultSolverAlg, using a type-stable sumtype approach as requested.

Key Features

• DefaultSDEAlgorithm(): Basic default algorithm that switches between SOSRI (non-stiff) and ImplicitRKMil (stiff)
• DefaultAdaptiveSDEAlgorithm(): Adaptive algorithm that selects solvers based on problem characteristics
• Automatic algorithm selection based on:
  • Stiffness detection
  • Noise type (additive, commutative, general)
  • Problem interpretation (Ito vs Stratonovich)
  • Mass matrix presence
• solve(prob, nothing) now uses the default algorithm automatically

Implementation Details

• Uses EnumX for defining solver choices (similar to OrdinaryDiffEq.jl)
• Leverages existing AutoAlgSwitch mechanism for automatic switching
• Type-stable implementation avoiding runtime dispatch
• Based on the algorithm selection logic from DifferentialEquations.jl's sde_default_alg.jl

Algorithm Mappings

The default algorithm selection follows this logic:

Problem Type	Non-stiff Algorithm	Stiff Algorithm
Default	SOSRI	ImplicitRKMil
Additive noise	SOSRA	ISSEM
Commutative noise	RKMilCommute	ImplicitRKMil
Stratonovich	Appropriate variant	ImplicitRKMil with Stratonovich

Tests

Added comprehensive test suite (test/default_alg_test.jl) covering:

• Basic default algorithm usage with solve(prob, nothing)
• Algorithm hints and adaptive selection
• Various problem types (scalar, system, stiff)
• Algorithm selection logic
• isdefaultalg checks

All tests pass successfully.

Example Usage

using StochasticDiffEq

# Simple SDE - will use default algorithm
f(u, p, t) = 1.01 * u
g(u, p, t) = 0.87 * u
prob = SDEProblem(f, g, 1.0, (0.0, 1.0))

# These all use the default algorithm
sol = solve(prob)  # or
sol = solve(prob, nothing)  # or
sol = solve(prob, DefaultSDEAlgorithm())

# Use hints for better algorithm selection
sol = solve(prob, DefaultAdaptiveSDEAlgorithm(alg_hints = [:stiff]))

Process Documentation

This implementation was created following the pattern from OrdinaryDiffEq.jl's DefaultODEAlgorithm:

1. Analyzed OrdinaryDiffEq.jl's type-stable sumtype approach
2. Studied DifferentialEquations.jl's sde_default_alg.jl for algorithm selection logic
3. Adapted the approach for StochasticDiffEq.jl's existing infrastructure
4. Used existing AutoAlgSwitch mechanism for type stability
5. Added EnumX dependency for solver choice enumeration
6. Formatted code with JuliaFormatter using SciMLStyle

🤖 Generated with Claude Code

Co-Authored-By: Claude [email protected]