Methods and Models Overview

Table of Contents

• 1. Governing Model Snapshot
• 2. Runtime Execution Order
• 3. Method Map
• 4. External Method References
• 5. How To Read These Pages
• 6. What Makes a Method Page Authoritative

Type ExplanationFor Readers evaluating the numericsStatus Current behavior

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.
• Fractional-Step (Projection) Method: predictor/projection incompressible update.
• Dual-Time Picard Jameson RK 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.
• LES Turbulence Closure: subgrid closure, the Germano-Lilly dynamic procedure, coefficient averaging, and limiting, the coefficient-free WALE and Vreman models, and grid filter widths.
• 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.
• Newton–Krylov Momentum Solver: matrix-free PETSc SNES/GMRES momentum solver and its deterministic-residual invariant.
• 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.

4. External Method References

For readers connecting PICurv implementation to the CURVIB literature, these are the primary starting references:

• Borazjani I, Ge L, Sotiropoulos F. "Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies." Journal of Computational Physics 227(16), 7587-7620 (2008). DOI: 10.1016/j.jcp.2008.04.024.
• Borazjani I, Di Achille P, D'Souza RM, et al. "The functional role of left atrial flow in ventricular filling and flow evolution in the left ventricle." Annals of Biomedical Engineering 41(6), 1265-1275 (2013). DOI: 10.1007/s10439-013-0758-9.

Project context:

• PICurv project page: https://vishalkandala.me/projects/Picurv/

5. How To Read These Pages

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.

6. What Makes a Method Page Authoritative

Each method page is a contract between users and maintainers, not a literature summary. Read it in this order:

1. Scope and assumptions — supported grid, boundary, solver, and restart conditions; unsupported combinations must be stated explicitly.
2. Discrete/runtime behavior — equation or algorithm, field locations, update order, and the C functions that implement it.
3. User controls — the YAML fields, defaults, generated PETSc options, and logs that show whether the method is behaving as configured.
4. Validation evidence — the unit, smoke, or regression checks that protect the stated behavior, including any limits on what has been exercised.
5. Extension boundary — which parser, data structure, runtime dispatch, and tests must change together when adding a model or selector.

A numerical claim is authoritative only when it can be traced to the current runtime path and a named validation surface. When this page and an older example, paper, or legacy option disagree, use the current configuration reference and runtime-specific method page, then correct the stale material.