Add/improve stage predictors for ESDIRK methods now in tableau form

[singhharsh1708]

Background

Following the SDIRK tableau-form unification (#3196, #3650, #3651), all SDIRK/ESDIRK methods route through ESDIRKIMEXCache + _perform_step_iip!/_oop!. The per-stage Newton initial guess ("predictor") is the initial value handed to nlsolve! for each stage z_i, currently selected in _build_initial_guess_efs_{true,false} from: const_stage_guess, the tableau α matrix, or zero.

Goal (per @ChrisRackauckas)

Make the full SUNDIALS ARKODE implicit-predictor menu available generically to every method, keep the tableau-derived custom α predictors as another option, and benchmark them all — do the custom ones actually win?

Predictor menu to implement

Selectable via a predictor option on the algorithms, wired into the generic stage-guess path:

• :trivial (SUNDIALS 0) — z_i = u_n (Δu = 0). This is exactly today's empty-α default. already effectively present.
• :max_order (SUNDIALS 1) — evaluate the previous step's dense-output interpolant at the stage time t_n + c_i·dt, at full available order. Building block already exists: current_extrapolant! / ode_extrapolant! (currently only used for stage 1 when extrapolant == :interpolant). Needs generalizing to all stages.
• :variable_order (SUNDIALS 2) — same interpolant but with order reduced as the stage abscissa c_i extrapolates further.
• :cutoff_order (SUNDIALS 3) — full order for stages with small c_i, drop to order 1 beyond a cutoff.
• :tableau — the existing custom α / const_stage_guess predictors (KenCarp/Kvaerno/TRBDF2/Cash4).

Variable/cutoff order need an order-limited interpolant evaluation, which the current dense-output code doesn't expose — that's the main new infrastructure piece.

Current state by method

The ESDIRK*L2SA family (ESDIRK54I8L2SA, ESDIRK436L2SA2, ESDIRK437L2SA, ESDIRK547L2SA2, ESDIRK659L2SA) ships with no predictor (empty α ⇒ trivial). KenCarp/Kvaerno/TRBDF2/Cash4 carry tuned α predictors.

Measured impact (reltol = abstol = 1e-8)

Even a naive copy-previous-stage predictor (α[i][i-1]=1) on the no-predictor ESDIRK*L2SA family cuts function evaluations measurably — confirming predictors matter here:

Method	Problem	nf (trivial)	nf (copy-prev)	Δ
ESDIRK54I8L2SA	VanDerPol	33515	28642	−14.5%
ESDIRK436L2SA2	VanDerPol	37624	33600	−10.7%
ESDIRK437L2SA	VanDerPol	38834	34834	−10.3%
ESDIRK54I8L2SA	Pollution	2457	2278	−7.3%
ESDIRK436L2SA2	Robertson	11063	9938	−10.2%

(Predictors also shift naccept indirectly via the nlsolve convergence-rate estimate feeding the controller, so this is a behavioral change requiring validation, not a free swap.)

Proposed work

• Add a generic predictor option (:trivial, :max_order, :variable_order, :cutoff_order, :tableau) wired into _build_initial_guess_efs_{true,false}.
• Generalize current_extrapolant!-based prediction to all stages (not just stage 1) for :max_order.
• Add order-limited interpolant evaluation to support :variable_order / :cutoff_order.
• Keep the tableau custom α predictors as :tableau.
• Benchmark all predictors × methods × stiff problems (nf / naccept / wall-time) — determine whether custom α beats the generic interpolant predictors, and pick sensible per-method defaults.
• Add an nf/naccept benchmark to guard predictor regressions.

cc @ChrisRackauckas — updated to reflect the SUNDIALS predictor menu scope.

[ChrisRackauckas]

which passes an empty α ⇒ stage initial guess is zero

This corresponds to guessing u_{n+1} = u_n, which isn't a bad guess but isn't particularly great. It is a good option to have available to all methods though.

[ChrisRackauckas]

https://sundials.readthedocs.io/en/latest/arkode/Mathematics_link.html#implicit-predictors all of these should be available as options, the one pointed to by putting z=0 (i.e. delta u = 0) is the trivial predictor. So we should do all of these as generic, plus custom predictors from the tableau. Then benchmark them all: are the custom ones even better?

[singhharsh1708]

Current mapping:

• :trivial (SUNDIALS 0) = today's empty-α default (Δu = 0)
• :max_order (SUNDIALS 1) can reuse current_extrapolant!; currently it only handles stage 1, so I’ll generalize it to arbitrary stage times t_n + c_i*dt
• :variable_order / :cutoff_order (SUNDIALS 2/3) need order-limited interpolant evaluation, which dense output doesn’t currently expose
• :tableau keeps the existing custom α predictors

The early ESDIRK*L2SA numbers already show predictors matter quite a bit (~7–15% nf reduction from a naive copy-previous-stage guess), so benchmarking all predictor strategies against the tuned tableau predictors should be interesting.

I’ll start by wiring the predictor option into the generic stage-guess path and implementing :trivial + :max_order first.