feat(Rosenbrock): post-solve per-process verbose diagnostics

[haakon-e]

What this adds

Adds post-solve per-process tendency diagnostics for the RosenbrockAverage modes (1M and 2M+P3) as a Verbose{M} wrapper around any mode, reporting the contribution of each microphysical process as realized by the implicit solve.

Why the decomposition is exact

Each substep solves a system that is linear in the right-hand side f:

(I/h − J) Δ = f,   f = Σₚ fₚ          (fₚ = the p-th process tendency)
⇒  Δ = Σₚ Δₚ   with   (I/h − J) Δₚ = fₚ

so the per-process increments Δₚ sum to the substep increment Δ. With nsub > 1 each substep is re-evaluated at the state it starts from, and the per-process increments and the clamp term are accumulated across substeps, so the relation holds for the averaged tendency. The positivity clamp max(x + Δ, 0) is the only nonlinear step; it is returned as a single correction term c that is not attributable to any one process:

Σₚ Δₚ + c = realized net increment.

The clamp is a no-op for the donor option (its rates are floored) and binds only rarely for the exact option, since the saturation adjustment already bounds growth. With the clamp disabled the decomposition is exactly Σₚ Δₚ = net, which is a reason it could be made optional.

Testing

Exactness tests confirm the per-process increments plus the clamp correction reconstruct the net to round-off (≈ 1e-13 in Float64) and that the verbose net equals the non-verbose net, for both schemes and precisions. The non-verbose path is unchanged and allocation-free. The verbose path currently allocates and is intended for offline diagnostics; an allocation-free version (a preallocated per-process accumulator, as in the earlier LinearizedAverage verbose variant) is a possible refinement.

[haakon-e]

This change is part of the following stack:

• feat(P3): Gauss-Legendre quadrature, integrand breakpoints -> GL(6) default #741
  • feat(Rosenbrock): unified RosenbrockAverage substepping framework + 2M+P3 exact-AD mode #726
    • feat(Rosenbrock): 1M donor / coupled / exact modes on the unified framework #731
      • feat(Rosenbrock): post-solve per-process verbose diagnostics #732 ◀
        • fix(Rosenbrock): saturation adjustment on max(S_ice, S_liq) #737
          • perf(BMT): recover the primal tendency from the Rosenbrock Jacobian pass #742
            • feat(BMT): analytic ManualJacobian for the 2M+P3 Rosenbrock substep #743
              • feat(BMT): coupled Tier-3 donor linearization for the 2M+P3 ManualJacobian #744

_{Change managed by git-spice.}