Numerical Stability

Goal

Provide a repeatable checklist and script-driven checks to keep time-dependent simulations stable and defensible.

Requirements

• Python 3.8+
• NumPy (for matrix_condition.py and von_neumann_analyzer.py)
• See scripts/requirements.txt for dependencies

Inputs to Gather

Input	Description	Example
Grid spacing dx	Spatial discretization	0.01 m
Time step dt	Temporal discretization	1e-4 s
Velocity v	Advection speed	1.0 m/s
Diffusivity D	Thermal/mass diffusivity	1e-5 m²/s
Reaction rate k	First-order rate constant	100 s⁻¹
Dimensions	1D, 2D, or 3D	2
Scheme type	Explicit or implicit	explicit

Decision Guidance

Choosing Explicit vs Implicit

Is the problem stiff (fast + slow dynamics)?
├── YES → Use implicit or IMEX scheme
│         └── Check conditioning with matrix_condition.py
└── NO → Is CFL/Fourier satisfied with reasonable dt?
    ├── YES → Use explicit scheme (cheaper per step)
    └── NO → Consider implicit or reduce dx

Stability Limit Quick Reference

Physics	Number	Explicit Limit (1D)