From one grid
to a hierarchy.

Choose a stage to inspect the difference. These are schematic 2D grids, not simulation results.

TRN-01 is a numerical effort on the existing grid. Start with current momentum budgets, test a conservative scalar building block, then derive MAC momentum transport and qualify boundaries and cut cells. A scalar test is not proof of momentum conservation or a delivered temperature feature.

Starting candidate: limited second-order finite-volume transport compatible with our MAC grid; exact method selection remains open. Scope, tests and acceptance. AMReX supports different equation formulations; conservation is our design choice, not a requirement of its data types.

Keep method selection small and explicit.

We already support SemiLagrangian and MacCormack. TRN-01 will retain useful choices and evaluate one conservative candidate. Select one method per run, use it across all patches and levels, and record it with the scientific settings. Each method owns its temporary buffers and declares its timestep, stencil and geometry requirements.

Support is qualified separately for each method and configuration. Initially S3 can require the accepted conservative method. Unsupported combinations must be rejected clearly; the solver must not silently switch methods.

Drill down: method selection and integration · current transport equations · shared-flux derivation.

Drill down: physical equations and MAC control volumes · why scalar transport is not enough for momentum.

S2 must use the accepted transport method's reconstruction, ghost reach and time stages, and apply a consistent flux across shared interfaces. Any retained semi-Lagrangian method also needs patch-aware departure-point sampling in place of current local clamping or wrapping.

Drill down: interface cancellation, average-down and refluxing · cut-cell and moving-boundary balances. References: AMReX boundary filling and AMReX-Hydro projection methods.

Both can use box descriptors. They need separate identities and ownership. Existing GpuMgHierarchy is the second kind.

Names that make the responsibilities clear

Separate GridMetrics' coordinate scalars from borrowed embedded-boundary metrics. Share a canonical field-placement description between runtime and output. A level coordinator schedules patch kernels; physical boundary application, same-level exchange and later coarse/fine filling have distinct jobs and explicit stage freshness.

Keep meaningful names such as SimulationSession, MacState, PressureWorkspace and GpuMgHierarchy. Naming, ownership and migration design.

Working simulation fields, repeated numerical geometry, transport, solvers and diagnostics belong on the device. Complete reductions and dependent coefficients, convergence evaluation and admissible timestep calculations there. Downloading whole fields or reduction partials to finish those calculations on the CPU is a migration gap, not an acceptable new implementation.

Gate: new or migrated numerical paths have no bulk field downloads or CPU completion of reductions. Audit direct and indirect copies, metadata transfer counts/bytes, synchronization time, launch overhead, peak memory and end-to-end performance. Count requested output separately. Scientific checks, precision and existing performance limits remain binding.

GPU-01 scope, source inventory and acceptance · Algorithm execution details.

The algorithm companion starts with the physical model and opens into discrete equations, source code and validation protocols. Current behavior, proposed algorithms and unresolved derivations are labeled separately.

1. Start with physics and field locationsIncompressible flow, units and staggered control volumes.
2. Trace today's complete time stepForces, diffusion, pressure signs and continuity.
3. Inspect today's transport equationsBacktracing, interpolation, correction and limiting.
4. Derive a conservative updateShared transfers and an illustrative scalar algorithm.
5. Identify the work still to deriveStaggered momentum, cut volumes and moving walls.
6. Review the proof and test obligationsConservation, stability, accuracy and physical validation.

Each effort follows the execution policy for the work it changes. AMR-02 does not require all of GPU-01 to finish; TRN-01, S2 and S3 are not blocked by unrelated migration gaps. New numerical paths must keep their work resident, document bounded host coordination and preserve the applicable scientific, memory and performance gates.

1. AMR-02

   Align today's structures

   Inventory consumers, introduce canonical descriptors, migrate ownership and callers, then remove superseded representations.

   Gate: existing inputs, fields, measurements, outputs and scientific acceptance remain valid. No new mesh capability, bulk field transfers or abstraction-driven performance/memory regression. Existing GPU-01 gaps remain explicitly tracked.

2. TRN-01

   Validate selectable single-grid transport

   Measure current budgets, test shared scalar fluxes, formulate MAC momentum transport, then qualify boundaries, cut cells and separately supported motion. Integrate one recorded method choice per run with explicit support, timestep and memory requirements.

   Gate: conservation and stability checks pass, retained methods preserve their scientific gates, and preselected cases demonstrate candidate accuracy benefit with acceptable cost. Selection is reproducible and unsupported configurations are rejected.

3. S2

   Connect patches at one resolution

   Begin with the same problem split in two. Extend the accepted transport method with shared fluxes and same-level sampling, plus coupled pressure/diffusion, geometry, measurements and output.

   Gate: partitioned and monolithic solutions and momentum budgets agree within declared tolerances, including periodic and cut interfaces. A simple two-patch test alone does not finish S2.

4. S3 / fixed

   Add resolution in one region

   Start with one coarse base, one aligned 2:1 fine patch and synchronized timesteps. Add coverage masks, transfers and composite equations.

   Gate: accuracy and conservation pass against uniform-fine references, with less measured memory or time at comparable measurement accuracy.

5. S3 / adaptive

   Let refined regions change

   Extend fixed refinement to multiple patches/levels. Then add GPU tagging and safe field transfer, with bounded compact metadata for host layout management and allocation, plus geometry/solver rebuilding.

   Gate: refine/derefine cycles preserve the required budgets and constraints; metadata transfers and regrid costs are measured. Thermal and moving-geometry combinations need their own evidence. Fully device-driven regridding and time subcycling are later decisions.

S2 layout work can overlap transport development. Its transport interfaces follow TRN-01's established method. Coordinate scalar work with T1, while keeping thermal acceptance separate.