NonlinearSolveAlg: let inner solver refresh J when no W is reused (fixes master nlstep order failures)

[ChrisRackauckas-Claude]

Please ignore until reviewed by @ChrisRackauckas. Draft opened by an agent.

Fixes two failing tests on master

test/modelingtoolkit/nlstep_tests.jl currently fails on master (17 pass / 2 fail / 6 broken): the FBDF + NSA(TrustRegion) Robertson order assertions at lines 60 and 116 measure order 0.6152 against bounds |𝒪est−1| < 0.3 / 0.35. (#3807 read this as "order 1.38" — it's 1 + 0.3848 in the test output; the order collapsed below 1.)

Root cause

#3703 changed both compute_step! methods from

recompute_jacobian = cache.W === nothing || (nlsolver.iter == 1 && cache.new_W)

to

recompute_jacobian = nlsolver.iter == 1 && (cache.W === nothing || cache.new_W)

which freezes the inner NonlinearSolve Jacobian after outer iteration 1 on the paths without a reusable W (MTK nlstep, DAE, OOP). The inner solve silently degrades to modified Newton while the outer η·ndz < κ termination keeps its calibration, so every step carries a dt-independent unconverged remainder. Measured l∞ errors over dts = 2^-15…2^-18: master [2.22e-7, 1.12e-7, 5.60e-8, 6.18e-8] (error stops halving, then rises); with this fix […, 1.38e-7, 5.60e-8, 2.80e-8] (clean halving). Warmed TrustRegion and NewtonRaphson give bit-identical trajectories — not TR-specific, and consistent with the principle that a converged inner solve must not change the ODE solution.

Fix

Restore the pre-#3703 condition: W === nothing paths let the inner solver refresh J every outer iteration; the W-reuse (modified-Newton) path is untouched — with a reused W the inner Jacobian is the ODE-side W by construction, so refreshing beyond iter 1 remains pointless there.

Verification (local)

• A/B on master: this exact change turns nlstep_tests.jl from 17 pass / 2 fail into 19 pass / 0 fail, with orders 1.2576 and 1.3035 back inside their bounds.
• newton_tests.jl + linear_nonlinear_tests.jl pass against this branch.

Related findings from the same investigation (separate issues)

• NonlinearSolveAlg(TrustRegion) + W-reuse: displacement-based ndz declares convergence on rejected TR steps → silently wrong solutions #3817: NSA(TrustRegion)+W-reuse can declare convergence from displacement-ndz on a rejected TR step → constant 4.7e-2 error across dts (not fixed by this PR).
• reinit! resets trust region radius to build-time initial value; recomputed u0_norm is dead code NonlinearSolve.jl#1004: reinit! resets the trust radius to its build-time value.

🤖 Generated with Claude Code

[ChrisRackauckas-Claude]

CI triage: the failing lanes here are master-baseline, not from this change — StochasticDiffEq QA/downgrade + GPU lanes fail on all concurrent PRs including the 2-line #3816; the OrdinaryDiffEqNonlinearSolve Core lane failure is the pre-existing Dependent Mass Matrix Tests error (mass_matrix_tests.jl:283, OOP NLNewton + non-concrete operator W, 221 pass/2 error/4 broken — the limitation #3796 documents), reproduced identically on unmodified master locally. The MTK nlstep lane failures are the two order-test failures that #3818 fixes.