Methods and Models Overview

Table of Contents

• 1. Governing Model Snapshot
• 2. Runtime Execution Order
• 3. Method Map

This section maps PICurv's numerical methods to the code paths that execute each step. It is intended as the bridge between theory-level terminology and what the current codebase actually does.

1. Governing Model Snapshot

PICurv advances incompressible flow in non-dimensional form on curvilinear grids:

\[ \frac{\partial \mathbf{u}}{\partial t} + \nabla\cdot(\mathbf{u}\mathbf{u}) = -\nabla p + \frac{1}{Re}\nabla^2\mathbf{u} + \mathbf{f}, \qquad \nabla\cdot\mathbf{u}=0. \]

Operationally, the solver uses a projection workflow:

1. build momentum residuals,
2. advance momentum with selected strategy,
3. solve Poisson equation for pressure correction,
4. project velocity to divergence-free space,
5. execute particle coupling (if enabled).

2. Runtime Execution Order

At runtime, the top-level sequence is:

1. CreateSimulationContext parses generated control files.
2. InitializeEulerianState sets initial Eulerian fields.
3. FlowSolver advances one fluid step (or AnalyticalSolutionEngine for analytical mode).
4. Particle stage (when active): interpolation -> motion -> relocation -> physics -> scatter.

The method pages below document each major stage in detail.

3. Method Map

• CurvIB Method Overview: curvilinear grid/metric framework and immersed-boundary context.
• Fractional-Step (Projection) Method: predictor/projection incompressible update.
• Dual-Time Picard RK4 Momentum Solver: implicit-in-physical-time momentum iteration.
• Pressure-Poisson, GMRES, and Multigrid: Poisson assembly and PETSc multigrid/KSP path.
• Walking Search for Particle Location: particle cell location and migration orchestration.
• Trilinear Interpolation and Particle-Grid Projection: Eulerian-Lagrangian field exchange.
• IEM Mixing and Statistical Averaging: particle micromixing and post statistics kernels.
• Momentum Solver Implementations: supported momentum-solver options and dispatch status.
• Analytical Solution Modes: analytical Eulerian modes and geometry policies.
• Initial Condition Modes: Eulerian and particle initialization modes.
• Particle Model and Coupling Overview: end-to-end particle model pipeline.
• Boundary Conditions Guide: boundary handler combinations and runtime enforcement path.
• Particle Initialization and Restart Guide: detailed particle seeding/restart/migration behavior.
• C Runtime Execution Map: code-level startup and timestep execution map.

1. General Usage

Use each page with two goals:

1. theory alignment (which equation/model is being approximated),
2. implementation alignment (which function actually executes it now).

When behavior differs from classical textbook formulations, the implementation notes take precedence for this repository.