Dual-Time Picard RK4 Momentum Solver

Table of Contents

• 1. Algorithmic Model
• 2. Convergence and Backtracking
• 4. Core Code Touchpoints

This page documents the currently active implicit-style momentum strategy in PICurv: dual-time Picard iteration with RK4 pseudo-time smoothing.

1. Algorithmic Model

The solver enforces the physical-time momentum equation by iterating a pseudo-time equation:

\[ \frac{\partial \mathbf{U}}{\partial \tau} = -\Big(R_{spatial}(\mathbf{U}) + R_{time}(\mathbf{U})\Big). \]

Implementation details from ComputeTotalResidual:

• R_spatial comes from ComputeRHS
• R_time uses BDF1/BDF2-style terms from Ucont, Ucont_o, and Ucont_rm1,
• RK4 stages use coefficients \(\{1/4,1/3,1/2,1\}\).

2. Convergence and Backtracking

Per pseudo-iteration, the solver tracks:

• \(\|\Delta U\|_\infty\),
• relative solution update \(\|\Delta U_k\|/\|\Delta U_0\|\),
• residual norms and growth ratios.

Backtracking is triggered when residual and update growth indicate divergence (or NaN), then:

1. state rolls back to the last accepted pseudo-state,
2. pseudo-CFL is reduced,
3. iteration retries from the restored state.

Pseudo-CFL is also adaptively ramped on successful steps and clamped by configured min/max bounds.

2. File-Grid Configuration

User-facing configuration (solver.yml) maps to:

• strategy.momentum_solver -> -mom_solver_type
• tolerances.max_iterations -> -mom_max_pseudo_steps
• tolerances.absolute_tol -> -mom_atol
• tolerances.relative_tol -> -mom_rtol
• tolerances.step_tol -> -imp_stol
• momentum_solver.dual_time_picard_rk4.pseudo_cfl.* -> pseudo-CFL flags
• rk4_residual_noise_allowance_factor -> -mom_dt_rk4_residual_norm_noise_allowance_factor

Parsing and normalization are performed in scripts/pic.flow, with final option ingestion in function CreateSimulationContext during setup.

4. Core Code Touchpoints

• implementation: MomentumSolver_DualTime_Picard_RK4
• explicit comparator path: MomentumSolver_Explicit_RungeKutta4
• runtime dispatch: FlowSolver
• options ingestion: CreateSimulationContext

4. What This Means For Users

Common stability tuning order:

1. reduce initial pseudo-CFL,
2. tighten/loosen residual-noise allowance slightly,
3. adjust pseudo-CFL growth/reduction factors,
4. revisit grid quality and boundary consistency if instability persists.

For many cases, robust Poisson settings and sane initialization matter as much as dual-time tolerances.