Fractional-Step (Projection) Method

Table of Contents

• 1. Discrete Method Skeleton
• 2. Code Path In PICurv
• 3. Boundary and Geometry Handling
• 4. Diagnostics To Watch
• 5. Related Pages
  • CFD Reader Guidance and Practical Use
    • What To Extract Before Changing A Case
    • Practical CFD Troubleshooting Pattern

PICurv uses a projection-style incompressible update: momentum predictor, pressure correction, velocity projection.

1. Discrete Method Skeleton

Given velocity \(\mathbf{u}^n\), the solver computes a provisional field \(\mathbf{u}^*\) from momentum terms, then solves pressure correction:

\[ \nabla^2 \phi = \frac{1}{\Delta t}\nabla\cdot\mathbf{u}^*, \]

followed by:

\[ \mathbf{u}^{n+1} = \mathbf{u}^* - \Delta t\nabla\phi, \qquad p^{n+1}=p^n+\phi. \]

In curvilinear form, gradient components use metric terms (ICsi/IEta/IZet, etc.) and Jacobian inverses in the projection kernel.

2. Code Path In PICurv

• stage orchestrator: FlowSolver
• momentum residual assembly: ComputeRHS
• pressure solve (MG/KSP): PoissonSolver_MG
• pressure update: UpdatePressure
• velocity projection: Projection

@ref Projection explicitly uses boundary-aware stencils and periodic-edge correction logic before final ghost updates and Contra2Cart conversion.

3. Boundary and Geometry Handling

Key implementation details:

• one-sided derivative stencils are used when centered stencils touch solids (Nvert checks),
• periodic edges are corrected in dedicated boundary loops,
• driven-periodic channel cases call the flux-profile corrector in projection path.

These are part of why projection behavior in PICurv differs from simplified textbook pseudocode.

4. Diagnostics To Watch

For projection health, monitor:

• continuity/divergence diagnostics after correction,
• Poisson KSP convergence history logs,
• step-to-step stability of pressure and corrected velocity.

Persistent divergence or noisy pressure correction usually indicates a Poisson tuning, grid-quality, or BC consistency issue.

5. Related Pages

• Dual-Time Picard RK4 Momentum Solver
• Pressure-Poisson, GMRES, and Multigrid
• Common Fatal Errors and Fixes

CFD Reader Guidance and Practical Use

This page describes Fractional-Step (Projection) Method within the PICurv workflow. For CFD users, the most reliable reading strategy is to map the page content to a concrete run decision: what is configured, what runtime stage it influences, and which diagnostics should confirm expected behavior.

Treat this page as both a conceptual reference and a runbook. If you are debugging, pair the method/procedure described here with monitor output, generated runtime artifacts under runs/<run_id>/config, and the associated solver/post logs so numerical intent and implementation behavior stay aligned.

What To Extract Before Changing A Case

• Identify which YAML role or runtime stage this page governs.
• List the primary control knobs (tolerances, cadence, paths, selectors, or mode flags).
• Record expected success indicators (convergence trend, artifact presence, or stable derived metrics).
• Record failure signals that require rollback or parameter isolation.

Practical CFD Troubleshooting Pattern

1. Reproduce the issue on a tiny case or narrow timestep window.
2. Change one control at a time and keep all other roles/configs fixed.
3. Validate generated artifacts and logs after each change before scaling up.
4. If behavior remains inconsistent, compare against a known-good baseline example and re-check grid/BC consistency.