[simone-silvestri]

This PR includes some nice implementations to improve order and accuracy of the split explicit barotropic solver:

• adds second order and third order filters (which have second and third order moments equal to zero)
• adds higher order barotropic stepping for the barotropic solver, required to achieve overall higher temporal order (RK3 and AB3)
• adds a slow forcing interpolation which is required for RK4 and improves the accuracy for RK3
• adds a strong stability preserving RK3 version coupled with the barotropic solver

[codecov]

Codecov Report

❌ Patch coverage is 71.08886% with 231 lines in your changes missing coverage. Please review.
✅ Project coverage is 74.35%. Comparing base (71effed) to head (2955347).
⚠️ Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
...SurfaceModels/hydrostatic_free_surface_ssp_step.jl	48.21%	58 Missing ⚠️
...dVerticalDiffusivities/time_step_catke_equation.jl	0.00%	52 Missing ⚠️
src/TimeSteppers/ssp_runge_kutta.jl	55.71%	31 Missing ⚠️
src/Advection/ghost_cell_reconstruction.jl	66.66%	16 Missing ⚠️
...itFreeSurfaces/step_split_explicit_free_surface.jl	92.48%	13 Missing ⚠️
src/Advection/immersed_advective_fluxes.jl	42.10%	11 Missing ⚠️
...itFreeSurfaces/split_explicit_averaging_kernels.jl	87.77%	11 Missing ⚠️
...xplicitFreeSurfaces/slow_forcing_reconstruction.jl	88.88%	7 Missing ⚠️
...xplicitFreeSurfaces/split_explicit_timesteppers.jl	68.18%	7 Missing ⚠️
...ceDissipationComputations/advective_dissipation.jl	72.72%	6 Missing ⚠️
... and 8 more

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #5812       +/-   ##
===========================================
+ Coverage   25.96%   74.35%   +48.38%     
===========================================
  Files         439      444        +5     
  Lines       27275    28255      +980     
===========================================
+ Hits         7083    21009    +13926     
+ Misses      20192     7246    -12946     

Flag	Coverage Δ
buildkite	70.21% <75.23%> (?)
distributed/output_combining	19.75% <12.65%> (-0.30%)	⬇️
distributed/tripolar	19.75% <12.65%> (-0.30%)	⬇️
julia	70.21% <75.23%> (?)
makie	20.20% <13.04%> (?)
metal	16.22% <10.71%> (-0.23%)	⬇️
reactant	20.20% <13.04%> (?)
sharding	20.20% <13.04%> (?)

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[Copilot]

Pull request overview

Improves split-explicit barotropic accuracy, filtering, topology handling, and Runge–Kutta integration.

Changes:

• Adds higher-order averaging kernels and slow-forcing reconstruction.
• Introduces SSPRK3 and higher-order barotropic substepping.
• Extends diagnostics, tripolar, multi-region, and implicit-advection support.

Reviewed changes

Copilot reviewed 33 out of 33 changed files in this pull request and generated 6 comments.

Show a summary per file

File	Description
test/test_tripolar_grid.jl	Updates serial-fold expectations.
test/test_split_explicit_free_surface_solver.jl	Tests kernels, reconstruction, and RK3 substeps.
test/test_implicit_diffusion_diagnostic.jl	Adds SSPRK3 coverage.
src/TurbulenceClosures/.../time_step_catke_equation.jl	Adds SSP CATKE stepping.
src/TimeSteppers/TimeSteppers.jl	Exports and registers new steppers.
src/TimeSteppers/ssp_runge_kutta.jl	Implements SSP Runge–Kutta.
src/TimeSteppers/split_runge_kutta.jl	Adds modified RK4 composition.
src/Operators/topology_aware_operators.jl	Adds tripolar-fold operators.
src/MultiRegion/multi_region_models.jl	Integrates new free-surface state.
src/MultiRegion/multi_region_grid.jl	Sizes multi-stage halos.
src/MultiRegion/cubed_sphere_grid.jl	Sizes cubed-sphere halos.
src/Models/VarianceDissipationComputations/VarianceDissipationComputations.jl	Adds scaling caches.
src/Models/VarianceDissipationComputations/update_fluxes.jl	Accumulates SSP fluxes.
src/Models/VarianceDissipationComputations/diffusive_dissipation.jl	Adds SSP diffusion accumulation.
src/Models/VarianceDissipationComputations/compute_dissipation.jl	Assembles SSP diagnostics.
src/Models/VarianceDissipationComputations/advective_dissipation.jl	Adds SSP advection accumulation.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/step_split_explicit_free_surface.jl	Refactors barotropic subcycling.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/SplitExplicitFreeSurfaces.jl	Exposes new solver APIs.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/split_explicit_timesteppers.jl	Implements barotropic RK3.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/split_explicit_free_surface.jl	Adds forcing and halo configuration.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/split_explicit_averaging_kernels.jl	Implements higher-order filters.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/slow_forcing_reconstruction.jl	Implements forcing interpolation.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/initialize_split_explicit_substepping.jl	Generalizes stage initialization.
src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/compute_slow_tendencies.jl	Caches stage forcing.
src/Models/HydrostaticFreeSurfaceModels/prescribed_hydrostatic_velocity_fields.jl	Adds SSP no-op dispatch.
src/Models/HydrostaticFreeSurfaceModels/implicit_free_surface.jl	Generalizes multi-stage dispatch.
src/Models/HydrostaticFreeSurfaceModels/HydrostaticFreeSurfaceModels.jl	Includes SSP implementation.
src/Models/HydrostaticFreeSurfaceModels/hydrostatic_free_surface_ssp_step.jl	Implements hydrostatic SSP stages.
src/Models/HydrostaticFreeSurfaceModels/hydrostatic_free_surface_model.jl	Registers and validates steppers.
src/Models/HydrostaticFreeSurfaceModels/explicit_free_surface.jl	Supports multi-stage stepping.
src/ImmersedBoundaries/mutable_immersed_grid.jl	Handles folded column depths.
src/Advection/implicit_vertical_advection.jl	Uses previous vertical spacing.
src/Advection/adaptive_implicit_vertical_advection.jl	Supports SSP stage durations.

Suppressed comments (1)

src/Models/HydrostaticFreeSurfaceModels/SplitExplicitFreeSurfaces/slow_forcing_reconstruction.jl:123

• This method ignores both ts.Nstages and the supplied Shu–Osher coefficients. Since SSPRungeKuttaTimeStepper publicly accepts arbitrary coefficient tuples, a four-stage reconstruction indexes past this three-element tuple, while other custom schemes receive incorrect sample nodes. Derive the stage abscissae from the recursion, or reject non-SSPRK3 coefficients before exposing reconstruction for them.

@inline stage_sample_times(::SSPRungeKuttaTimeStepper) = (0.0, 1.0, 1/2)

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[aklocker42]

Hi Simone, I ran into a z★ tracer-constancy problem at open boundaries, and this PR fixes part of it, so flagging it here.

Setup: c ≡ 1, ZStarCoordinate, SplitRungeKutta3, open north/south boundaries with GravityWaveRadiationBoundaryCondition carrying a zero-net inflow. On main, c drifts by ~2e-5 within 200 steps.

Cause: σ is advanced with η, but w̃ comes from Ũ/Ṽ, and the two disagree at the boundary faces:

1. Ũ/Ṽ are accumulated from U/V before the substep halo fill and transport pin, so the face at index 1 gets pre-fill values and the face at N+1 is never accumulated.
2. After substepping, fill_halo_regions!((Ũ, Ṽ)) overwrites the boundary faces with the open-boundary value.

This PR fixes (1) for the faces inside the kernel range by accumulating U★/V★ in the η kernel. On its own, though, the result is bit-identical to main, because (2) overwrites the boundary faces anyway. With both fixed (also accumulate the N+1 face on a closed right side, and give Ũ/Ṽ boundary conditions that skip normal-flow sides), c − 1 drops to ~3e-15.

I have a minimal version of both on top of main, with a test that fails on main (|c − 1| ≈ 7e-6) and passes with the fix. I'll open it as a draft PR once the GPU tests pass. Since it touches the same kernel, I'm happy to fold it into this PR instead if you'd rather.

(Under AB2 there is a separate, larger z★ non-constancy, ~3e-4 in the same setup, that this doesn't address.)