Newton-Krylov integrators default to Hegedüs warm start (require LinearSolve 5.1)

[ChrisRackauckas-Claude]

Important

Draft — please ignore until reviewed by @ChrisRackauckas.

Companion to SciML/LinearSolve.jl#1100 (now merged and released as LinearSolve v5.1.0), which added WarmStart + the WarmStart.Auto default to KrylovJL. This makes the integrator-aware default unconditionally live on LinearSolve 5.1.

What

OrdinaryDiffEqDifferentiation.default_krylov_warm_start(linsolve) resolves a Krylov solver left at LinearSolve.WarmStart.Auto (the KrylovJL default) to WarmStart.Hegedus, applied on the Newton nonlinear-solver path (NLNewton/NonlinearSolveAlg). Implicit integrators using KrylovJL_GMRES()/KrylovJL_FGMRES() now reuse the previous solution (Hegedüs-scaled) across the correlated linear solves of a Newton iteration.

• Rosenbrock / W-methods untouched: they never call the resolver, so their Auto solver stays a cold start (warm starting is unsafe there — no outer Newton iteration absorbs the within-tolerance stage-solve perturbation; benchmarks showed 7–13× slowdowns and step-rejection loops).
• Explicit choices respected: WarmStart.None/Previous/Hegedus are never overridden — only Auto is resolved.

Requires LinearSolve 5.1 (no guard, no dormancy)

WarmStart is a LinearSolve 5.1 feature. Rather than guard on isdefined(LinearSolve, :WarmStart) and be silently dormant on older LinearSolve, the WarmStart-using libraries now require LinearSolve = "5.1" and the guard is removed, so the default is always live. Every other monorepo sublibrary that declared a LinearSolve compat is moved to "5.1" as well: they all depend transitively on OrdinaryDiffEqNonlinearSolve/Differentiation, so they can never resolve with LinearSolve < 5.1 anyway — keeping a 3/4 lower bound would be dead metadata and would re-break the Downgrade CI (which floors deps to their minimum). Test sub-environments updated to match. This is effectively the OrdinaryDiffEq LinearSolve-5 migration, done here because the require-5.1 floor depends on it.

Minor version bumps for each sublibrary whose compat changed.

Verification (local, LinearSolve 5.1.0)

• Fresh resolution pins LinearSolve 5.1.0 with the two libraries requiring "5.1" — the registered NonlinearSolveBase/DiffEqBase/etc. chain already allows LinearSolve 5, so nothing caps it at 4.
• Resolver unit test (test/warm_start_default_tests.jl, Differentiation Core group): 7/7 pass — Auto→Hegedus, explicit values preserved, non-Krylov unchanged, other fields survive remake.
• FBDF(linsolve = KrylovJL_GMRES()) and Rodas5P(linsolve = KrylovJL_GMRES()) both solve Success; the resolver returns Hegedus for Auto with the guard removed.
• Earlier engagement check (ILU Brusselator): FBDF fires warm start (433 engagements), Rodas5P does not (0), explicit None on FBDF is cold (0), explicit Hegedus on Rodas5P is honored (1528).

CI validates the full umbrella resolution + all sublibrary test groups on LinearSolve 5.1.

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[ChrisRackauckas-Claude]

CI note: the downgrade-sublibraries failures (Differentiation, FIRK) are pre-existing on master — the Downgrade/Downgrade Sublibraries workflows are red across recent master commits (d2577da, 4b3cafd, 754b504), unrelated to this PR. This change keeps the same LinearSolve compat floor (3.75.0), so it doesn't affect downgrade resolution, and both the resolver and its test are guarded on isdefined(LinearSolve, :WarmStart) so they degrade to a no-op on LinearSolve 3.75. The failure is the known julia-downgrade-compat-vs-path-[sources] resolution issue, not a real floor incompatibility. (Triaging the master failure separately.)

The checks that actually exercise this PR — the OrdinaryDiffEqDifferentiation/OrdinaryDiffEqNonlinearSolve sublib test groups and Runic — are the ones to watch.

🤖 Generated with Claude Code

[ChrisRackauckas-Claude]

CI verdict — the checks that exercise this PR are green:

• Runic (format + suggestions): pass
• OrdinaryDiffEqDifferentiation Core tests (Julia 1 / lts / pre): pass — these run the new warm_start_default_tests.jl
• OrdinaryDiffEqNonlinearSolve Core: pending (same class)

Remaining red checks are all pre-existing on master, not introduced here:

• QA (Differentiation, NonlinearSolve): the SciMLTesting public API is rendered in docs check flags pre-existing public-but-undocumented internal helpers (wrapprecs, dolinsolve, build_J_W, update_W!, …). Confirmed failing on master (sublibrary run 30005108996). The new default_krylov_warm_start is deliberately not public, so it is not in the flagged list and adds nothing to this debt.
• downgrade-sublibraries: chronic (never green on master, tracked under Downgrade CI tests have been failing since workflow was added #2972/Re-enable downgrade tests #2977). This PR fixes the two floor conflicts that live in the files it touches (LinearSolve 3.75→4.3, SciMLOperators 1.24.3→1.24.4); any remaining transitive floor conflict is the broader monorepo cleanup.

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