Add support for DAE models with PathSim solvers

[DavidMStraub]

PyBaMM v26.7.0 was released today and it contains a breaking change: all models are now DAEs. (outdated)

PathSim v0.23.0 added support for DAEs.

This makes DAE support no longer a nice to have, but a prerequisite.

[DavidMStraub]

https://github.com/pybamm-team/PyBaMM/releases/tag/pybamm-v26.7.1.0 😒

[DavidMStraub]

The issue description is outdated, but the feature request still useful: using PathSim's internal DAE solver, we can run even the DFN model without cosimulation.

@milanofthe I launched an investigation into implementing this - it's straightforward, but needs a change in PathSim core. I could open a PR for it.

Claude:

---

Upstream requirements for running PyBaMM DAE models (DFN, lead_acid.Full) with PathSim solvers

A prototype based on SemiExplicitDAE runs DFN, DFN with lumped thermal, MPM, NewmanTobias and lead_acid.Full with ESDIRK43 and matches PyBaMM/IDAKLU to within 0.05 mV. It only works by overriding non-public parts of SemiExplicitDAE, so two additions to its public API are needed:

1. Analytic reduced Jacobian (required for all DAE models). Accept optional partial Jacobians (∂f_dyn/∂x, ∂f_dyn/∂z, ∂f_alg/∂x) and form the reduced Jacobian from them instead of using finite differences through the constraint solve. With finite differences, a DFN run covered only 2.6 s of simulated time in 150 s of wall time. With the analytic Jacobian it covers 600 s in 9–29 s.
2. Configurable tolerance for the algebraic solve (required for lead_acid.Full). The fixed absolute tolerance of 1e-9 cannot be reached by badly conditioned models: in lead_acid.Full the residual stalls at about 2e-7, so every solve runs to the iteration limit. Scaling the residual inside func_alg works around this, but the scale factor is model-dependent.

Optional: reusing the Jacobian across Newton iterations within a stage would make DFN about 3× faster, but it isn't required.

[milanofthe]

Go ahead with the PR in pathsim we can discuss details there.