GlobalDiffEq: add GlobalDefectCorrection global error estimation and control#3957
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This was referenced Jul 18, 2026
Carries SciML/GlobalDiffEq.jl#56 (GlobalAdjoint, adjoint_error_estimate; Cao and Petzold 2004, closes SciML/GlobalDiffEq.jl#4) into the monorepo, restructured as the package extension GlobalDiffEqSciMLSensitivityExt on SciMLSensitivity + QuadGK weakdeps. sensealg defaults to nothing and resolves to QuadratureAdjoint(autojacvec = true) when the extension loads; solving without the extension raises an instructive error. Public API and semantics otherwise match the original PR. Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
Implements the explicit general linear methods with built-in global error estimation from Constantinescu (2016), SIAM J. Numer. Anal. 54(6) (arXiv:1503.05166; PETSc's TSGLEE23/TSGLEE24/TSGLEE35), addressing SciML/GlobalDiffEq.jl#6 and SciML/GlobalDiffEq.jl#22, as proper OrdinaryDiffEqCore algorithms. The methods propagate the partitioned state (y, ε) where ε is an asymptotically correct global error estimate; solve/init on plain ODEProblems transparently extend to the ArrayPartition form and global_error_estimate(sol) extracts the estimates. The per-step ε increment is an asymptotically correct local error estimate driving standard step-size adaptivity. The paper's method A4 is defective as printed (violates b·c = 1/2, converges at order 1, absent from PETSc) and is deliberately not implemented. Verified locally on the unstable Prince42 problem: solution orders 2/2/3, estimate accuracy orders 3/3/4, est/true-error ratio 1.00. Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
Adds the DOPRI5-based order-5 scheme with cheap global error estimation of Makazaga and Murua (BIT 43, 2003) as another tableau in the GLEE general linear framework (SciML/GlobalDiffEq.jl#6, the coefficient- complete published representative of the Dormand-Duckers-Prince RK triple family). Stages 1-7 are the standard DOPRI5 stages, three extra stages propagate an order-6 companion solution, and a generic solution-stage FSAL detection reuses stage 7's evaluation for fsallast, giving exactly 9 function evaluations per step (verified by an nf accounting test). The tableau was verified with exact rational arithmetic against the order and independency conditions. Verified locally on the unstable Prince42 problem: solution order 5, estimate accuracy order 6, est/true-error ratio 1.04, 9.01 f-evals/step. Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
Implements the linearized error-transport estimator (SciML/GlobalDiffEq.jl#5; Shampine 1986 / Berzins 1988 / Lang-Verwer 2007): after a dense forward solve, the companion linear ODE ε' = Jε + d(t) driven by the dense-output defect d(t) = f(P(t)) - P'(t) is integrated for the endpoint global error, with matrix-free Jacobian-vector products through DifferentiationInterface (autodiff accepts any ADTypes backend, default AutoForwardDiff). gtol-controlled solves tighten local tolerances via the shared refinement loop in companion.jl, mirroring GlobalAdjoint. global_error_estimate(prob, alg) is the standalone estimator, also added for GlobalAdjoint as an alias of adjoint_error_estimate. Verified locally on Lotka-Volterra vs a 1e-13 reference: estimate to true-error ratios 0.995-0.999 at tolerances 1e-4/1e-6, and gtol=1e-7 control yields 9.8e-9 final error. Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
Implements solving-for-the-correction (SciML/GlobalDiffEq.jl#3 family; Zadunaisky 1976, Dormand-Duckers-Prince 1984, Dormand-Lockyer- McGorrigan-Prince 1989, validity under adaptive stepping by Calvo-Higham-Montijano-Randez 1996): the nonlinear companion ODE ε' = f(P + ε) - P' driven by the dense-output defect, requiring no Jacobian. Same estimator and gtol-control API as GlobalErrorTransport via the shared companion machinery. Verified locally on Lotka-Volterra vs a 1e-13 reference: estimate to true-error ratios 0.998-1.000 at tolerances 1e-4/1e-6, and gtol=1e-7 control yields 9.9e-9 final error. Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
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Ignore this PR until it has been reviewed by @ChrisRackauckas.
Stacked on #3956 (transport) — the tip of the remaining GlobalDiffEq series: adjoint (#3953) → GLEE (#3954) → MM5GEE (#3955) → transport (#3956) → this. #3929 (repo transfer) has merged and this branch is rebased onto
master; only the top commit is new here. Merge the base PRs first.Summary
Adds
GlobalDefectCorrection— global error estimation andgtol-based control by solving for the correction (defect correction), addressing the SciML/GlobalDiffEq.jl#3 method family (Zadunaisky 1976; Dormand–Duckers–Prince 1984; the per-step dense-output form of Dormand–Lockyer–McGorrigan–Prince 1989; asymptotic validity under adaptive stepping by Calvo–Higham–Montijano–Rández 1996).P(t), the nonlinear companion ODEε' = f(P(t) + ε) − P'(t)is integrated fromε(t₀) = 0;‖ε(T)‖₂estimates the endpoint global error. UnlikeGlobalErrorTransportthis needs no Jacobian, onlyfevaluations on perturbed arguments, and it is the generic coefficient-free realization of the Dormand–Prince triple idea for any dense-output solver.global_error_estimate(prob, alg; ...), andgtol-controlledsolvevia the shared refinement loop fromsrc/companion.jl.This completes the stacked series; after it merges, SciML/GlobalDiffEq.jl can be archived pointing at the monorepo, and SciML/GlobalDiffEq.jl#56 closed. Of the issue tracker, SciML/GlobalDiffEq.jl#2 (multistep GEE) is intentionally left unimplemented: the referenced Farzi (IJNAO 2016) construction reads the global error of a k-step LMM block from the Butcher-tree truncation error of a composite RK method, which requires the elementary differentials of
f— impractical as a general-purpose runtime estimator.Version 1.7.0.
Validation (run locally)
ODEDIFFEQ_TEST_GROUP=Core julia --project=lib/GlobalDiffEq -e 'using Pkg; Pkg.test()': pass — Basic 1/1, traits 3/3, Adjoint 14/14, GLEE 37/37, Companion estimators 28/28, BigFloat 2/2ODEDIFFEQ_TEST_GROUP=QA julia --project=lib/GlobalDiffEq -e 'using Pkg; Pkg.test()': pass — 22 passed, 1 pre-existing declared brokengtol = 1e-7control delivered a true endpoint error of 9.9e-9; argument-validation paths tested.🤖 Generated with Claude Code
https://claude.ai/code/session_01MfUh8tgp3iBmMv9nkBqoyY