Use Krylov for matrix-free split ExpRK operators

[AshtonSBradley]

Summary

Teaches fixed-step exponential RK methods to use the existing Krylov path when a SplitODEProblem linear operator is matrix-free.

This builds on the has_concretization support merged in SciMLOperators PR: SciML/SciMLOperators.jl#361

Fixes #2078.

Motivation

Operator-splitting problems using SciMLOperators matrix-free compositions can fail during cache construction because the non-Krylov path eagerly calls convert(AbstractMatrix, A).

Changes

• During prepare_alg, detect split linear operators with !has_concretization(A).
• Automatically reconstruct the same algorithm with krylov = true for LawsonEuler, NorsettEuler, ETDRK2, ETDRK3, ETDRK4, and HochOst4.
• Preserve m, iop, autodiff, and concrete_jac.
• Add a regression test for a matrix-free FunctionOperator \ DiagonalOperator * FunctionOperator split.

Type stability / allocations

The new check happens once during algorithm preparation. It does not add per-step work. Matrix-backed operators keep the existing dense cached path. Matrix-free operators move onto the already-existing Krylov path, avoiding failed matrix materialization.

Tests

On current master, with SciMLOperators v1.18.0:

julia --project=lib/OrdinaryDiffEqExponentialRK -e 'using OrdinaryDiffEqExponentialRK, LinearAlgebra, Test; using SciMLBase: successful_retcode; u0 = ComplexF64[1.0 + 0.5im, -0.5 + 0.25im, 0.75 - 0.125im, -0.25 - 0.5im]; λ = ComplexF64[-1.0, -2.0, -3.0, -4.0]; F = FunctionOperator((v, u, p, t) -> v, u0; T = ComplexF64, islinear = true, op_inverse = (v, u, p, t) -> v, opnorm = (_ -> 1.0)); L = cache_operator(F \ DiagonalOperator(λ) * F, u0); @test !has_concretization(L); prob = SplitODEProblem(L, (u, p, t) -> zero(u), u0, (0.0, 0.1)); sol = solve(prob, ETDRK4(), dt = 0.01, save_everystep = false); @test successful_retcode(sol); @test sol.alg.krylov; @test sol.u[end] ≈ exp.(0.1 .* λ) .* u0 rtol = 1.0e-6'

The OrdinaryDiffEq matrix-free split tests for this change pass on the rebased branch.

Pkg.test() for OrdinaryDiffEqExponentialRK was also rerun on Julia 1.12.5 after the rebase. The new Matrix-free SciMLOperator split test passes; the run still reports the existing ExpRK Autodiff error, which is independent of this change and left out of scope for this PR.

[AshtonSBradley]

Codex was used to help prepare and validate this PR.