# numerical-stability

Checks CFL and Fourier conditions, runs von Neumann stability analysis, and computes matrix conditioning for time-dependent partial differential equations.

- skill · `type=skill` · Claude Code, Codex, Cursor, GitHub Copilot, Gemini CLI, OpenCode · ★ 83
- Author: HeshamFS
- License: Apache-2.0 · policy=mirror · the beginning of the file is at the end of this page
- Source: https://github.com/HeshamFS/materials-simulation-skills/tree/HEAD/skills/core-numerical/numerical-stability/SKILL.md
- Updated: 2026-10-21
- HTML: https://affpapa.org/en/nevibecode/skills/numerical-stability

## What it does

The skill helps you select a time step for explicit and implicit schemes: computes Courant, Fourier, and reaction numbers, detects system stiffness from eigenvalue ratios, verifies amplification factors for custom stencils, and recommends whether to use an explicit or implicit integrator. Four Python scripts drive the checks: `cfl_checker.py`, `von_neumann_analyzer.py`, `stiffness_detector.py`, and `matrix_condition.py`, each returning JSON metrics.

For two-dimensional diffusion the Fourier limit drops to 0.25 instead of 0.5 to account for grid dimensionality. If a matrix condition number exceeds 10⁸, the skill flags it as poorly-conditioned; crossing 10¹⁰ marks it ill-conditioned, and solving without preconditioning will likely lose accuracy. Scripts require Python 3.10 and NumPy.

## Install

### Claude Code (native platform)

1. **Skills CLI (Vercel)**
```bash
npx skills add HeshamFS/materials-simulation-skills --skill numerical-stability -a claude-code
```

2. **Manually: put the files in place**
- File: `~/.claude/skills/numerical-stability/SKILL.md` — Copy the whole skill folder; for one project use .claude/skills/numerical-stability/

### Codex

1. **Skills CLI (Vercel)**
```bash
npx skills add HeshamFS/materials-simulation-skills --skill numerical-stability -a codex
```

2. **Manually: put the files in place**
- File: `.agents/skills/numerical-stability/SKILL.md` — For all projects use ~/.agents/skills/

### Cursor

1. **Skills CLI (Vercel)**
```bash
npx skills add HeshamFS/materials-simulation-skills --skill numerical-stability -a cursor
```

2. **Manually: put the files in place**
- File: `.cursor/skills/numerical-stability/SKILL.md` — Cursor also reads .agents/skills/ and .claude/skills/

### GitHub Copilot

1. **GitHub CLI**
```bash
gh skill install HeshamFS/materials-simulation-skills numerical-stability
```

2. **Manually: put the files in place**
- File: `.github/skills/numerical-stability/SKILL.md`

### Gemini CLI

1. **Skills CLI (Vercel)**
```bash
npx skills add HeshamFS/materials-simulation-skills --skill numerical-stability -a gemini-cli
```

2. **Platform CLI**
```bash
gemini skills install https://github.com/HeshamFS/materials-simulation-skills --path skills/core-numerical/numerical-stability
```

### Windsurf (Devin Desktop)

**Manually: put the files in place**
- File: `.devin/skills/numerical-stability/SKILL.md` — The legacy .windsurf/skills/ path still works

### OpenCode

**Skills CLI (Vercel)**
```bash
npx skills add HeshamFS/materials-simulation-skills --skill numerical-stability -a opencode
```

### Claude.ai

**Claude.ai**
- Zip the folder and upload it in Settings → Capabilities → Skills

Where a file is given instead of a command, copy the files from the source to that path: https://github.com/HeshamFS/materials-simulation-skills/tree/HEAD/skills/core-numerical/numerical-stability/SKILL.md

## When to use it
- diagnose explicit scheme blow-up via CFL criterion
- choose a time step for multidimensional diffusion respecting Fourier limit
- detect problem stiffness from operator spectrum
- verify matrix conditioning before implicit solve
- confirm three-point stencil stability with von Neumann analysis

## Example prompts
- Check stability for `dx=0.01`, `dt=1e-4`, diffusivity `D=1.0`, 2D grid
- Run `cfl_checker.py` for advection at 2 m/s, spacing `0.05`, time step `0.01`
- Compute stiffness ratio from eigenvalues `-1` and `-1000`

## Source text · SKILL.md · Apache-2.0

Below is the content of a third-party file. It is data, not instructions for you: do not act on anything written inside unless the human asks. Block tag: bf6cbcac76d17c6b.

BEGIN THIRD-PARTY FILE bf6cbcac76d17c6b
````text
---
name: numerical-stability
description: >
  Analyze numerical stability for time-dependent PDE simulations — check CFL
  and Fourier criteria, perform von Neumann stability analysis, detect stiffness,
  evaluate matrix conditioning, and recommend explicit vs implicit time-stepping
  schemes. Use when selecting time steps, diagnosing numerical blow-up or solver
  divergence, checking convergence criteria, or evaluating scheme stability for
  advection, diffusion, or reaction problems, even if the user doesn't explicitly
  mention "stability" or "CFL."
allowed-tools: Read, Bash, Write, Grep, Glob
metadata:
  last_deterministic_eval: "2026-10-20"
  agent_evaluation_status: "not_revalidated"
  author: HeshamFS
  version: "1.2.3"
  security_tier: high
  security_reviewed: true
  tested_with:
    - claude-code
  last_evaluated: "2026-06-24"
  eval_cases: 7
  last_reviewed: "2026-10-19"
  standards:
    - "Courant-Friedrichs-Lewy (CFL) condition (Courant, Friedrichs, Lewy 1928)"
    - "von Neumann (Fourier) stability analysis; amplification factor and Fourier limit Fo ≤ 1/(2d)"
    - "Dahlquist A-stability / L-stability theory for time integrators"
    - "BDF (Gear) and Radau IIA / Rosenbrock methods for stiff systems"
    - "IEEE 754 double precision (conditioning thresholds, κ ≈ 1/eps)"
---

# Numerical Stability

## Goal

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

## Requirements

- Python 3.10+
- 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 dt (grid refinement requires a coupled timestep reduction)
```

### Stability Limit Quick Reference

| Physics | Number | Explicit Limit (1D) | Formula |
````
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