NonlinearSolveAlg(TrustRegion) + W-reuse: displacement-based ndz declares convergence on rejected TR steps → silently wrong solutions

[ChrisRackauckas-Claude]

Found while investigating the master nlstep_tests.jl FBDF+NSA(TrustRegion) order failures (see #3807 discussion; the order-test failure itself is fixed separately).

Symptom

Fixed-dt FBDF + NonlinearSolveAlg(TrustRegion()) on Robertson (analytic jac, plain W-reuse path, no MTK nlstep) produces a final-state error of 4.741e-2, constant across dt = 2^-15 … 2^-18 — a dt-independent error floor, i.e. silently wrong results with Success retcodes. Stats show exactly 1 inner iteration and 1 f-eval per step.

Mechanism

compute_step! for NonlinearSolveAlg computes the convergence norm from the displacement: ndz = internalnorm(calculate_residuals(z .- nlcache.u, ...)) (newton.jl, from #3693). When TrustRegion rejects or truncates its internal step, nlcache.u barely moves, so ndz ≈ 0 — and the iter-1 η-convergence shortcut (iter == 1 && ndz < 1e-5, plus the first-iter trust gate enabled for cache.W !== nothing in nlsolve.jl) declares the stage converged with essentially no correction applied. A rejected globalization step reads as "converged" to the outer loop.

Two related consequences observed with the same mechanism:

• The θ≈1 stall detection (eps_around_one branch in nlsolve.jl) misfires for globalized inner solvers: a rejected TR step leaves fu unchanged → θ = 1 exactly → premature Divergence (after 2 iterations) or premature convergence depending on ndz.
• On a freshly built MTK nlstep problem, FBDF + NSA(TrustRegion) at dt=2^-15 aborts at step 0 with ConvergenceFailure after 2 inner iterations via that θ≈1 path (test files only pass because earlier testsets warm the shared nlprob; see also the companion NonlinearSolve.jl issue about reinit! resetting the trust radius to its build-time value).

Proposed fix direction

Outer convergence must not be declared from displacement-ndz when the inner solver did not actually accept a step / make progress: consult the inner cache (descent success / step acceptance / retcode) before treating a small ndz as convergence — or, longer-term, drive NonlinearSolveAlg inner solvers by their own residual-based termination instead of NLNewton's θ/η heuristics, which assume a plain (modified-)Newton contraction.

Repro scripts from the investigation are available; error table:

dt	final-state l∞ error (FBDF+NSA(TrustRegion), fixed dt)
2^-15	4.741e-2
2^-16	4.741e-2
2^-17	4.741e-2
2^-18	4.741e-2

(For comparison, FBDF+NSA(NewtonRaphson) and FBDF+NLNewton halve cleanly.)

🤖 Filed by an agent after local reproduction on master 72706ab, Julia 1.12.6, NonlinearSolve 4.20.1.