# differentiation-schemes

Choose a numerical differentiation method, generate finite-difference stencils, and estimate truncation error. Useful when discretizing PDEs or ODEs.

- 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/differentiation-schemes/SKILL.md
- Updated: 2026-10-21
- HTML: https://affpapa.org/en/nevibecode/skills/differentiation-schemes

## What it does

The skill relates the solution’s smoothness, boundary conditions, and target accuracy to a suitable scheme. It covers central, one-sided, compact, and spectral methods; for discontinuities, it points to upwind, TVD, or WENO approaches. Bounded domains need one-sided boundary stencils when using central differences.

Three scripts handle scheme selection, stencil generation, and error estimates: `scripts/scheme_selector.py`, `scripts/stencil_generator.py`, and `scripts/truncation_error.py`. They can return JSON. Python 3.10+ and NumPy are required for stencil computations, so the generator depends on NumPy. The workflow recommends checking results against manufactured solutions or through grid refinement.

## Install

### Claude Code (native platform)

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

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

### Codex

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

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

### Cursor

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

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

### GitHub Copilot

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

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

### Gemini CLI

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

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

### Windsurf (Devin Desktop)

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

### OpenCode

**Skills CLI (Vercel)**
```bash
npx skills add HeshamFS/materials-simulation-skills --skill differentiation-schemes -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/differentiation-schemes/SKILL.md

## When to use it
- choose a scheme for an advection- or diffusion-dominated problem
- generate a finite-difference stencil at a target accuracy
- create one-sided stencils for boundary points
- estimate truncation error scaling
- compare candidate schemes by dispersion and dissipation

## Example prompts
- Choose a scheme for a smooth periodic problem with a first derivative and 4th-order accuracy.
- Generate a central stencil for a second derivative with 4th-order accuracy.
- Estimate truncation error with `dx` 0.01, accuracy 2, and scale 1.0.

## 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: f835a971bc0f7408.

BEGIN THIRD-PARTY FILE f835a971bc0f7408
````text
---
name: differentiation-schemes
description: >
  Select and apply numerical differentiation schemes for PDE and ODE
  discretization — generate finite-difference stencils at arbitrary order and
  accuracy, choose between central, upwind, compact (Pade), and spectral
  methods, handle boundary stencils, and estimate truncation error scaling.
  Use when discretizing spatial derivatives, picking a scheme for advection-
  or diffusion-dominated problems, building custom stencils for nonstandard
  operators, or comparing dispersion and dissipation properties of candidate
  schemes, even if the user just says "how do I approximate this derivative"
  or "my solution is too diffusive."
allowed-tools: Read, Bash, Write, Grep, Glob
metadata:
  last_deterministic_eval: "2026-10-20"
  agent_evaluation_status: "not_revalidated"
  author: HeshamFS
  version: "1.2.2"
  security_tier: high
  security_reviewed: true
  tested_with:
    - claude-code
  last_evaluated: "2026-06-24"
  eval_cases: 4
  last_reviewed: "2026-06-23"
  standards:
    - "Fornberg (1988), Generation of finite difference formulas on arbitrarily spaced grids"
    - "Taylor-series truncation-error analysis / order-of-accuracy theory"
    - "Lele (1992), Compact finite difference schemes with spectral-like resolution"
    - "Richardson extrapolation and grid-convergence (observed-order) verification"
    - "Jiang & Shu (1996), ENO/WENO essentially non-oscillatory schemes"
---

# Differentiation Schemes

## Goal

Provide a reliable workflow to select a differentiation scheme, generate stencils, and assess accuracy for simulation discretization.

## Requirements

- Python 3.10+
- NumPy (for stencil computations)
- No heavy dependencies

## Inputs to Gather

| Input | Description | Example |
|-------|-------------|---------|
| Derivative order | First, second, etc. | `1` or `2` |
| Target accuracy | Order of truncation error | `2` or `4` |
| Grid type | Uniform, nonuniform | `uniform` |
| Boundary type | Periodic, Dirichlet, Neumann | `periodic` |
| Smoothness | Smooth or discontinuous | `smooth` |

## Decision Guidance

### Scheme Selection Flowchart

```
Is the field smooth?
├── YES → Is domain periodic?
│   ├── YES → Use central differences or spectral
│   └── NO → Use central interior + one-sided at boundaries
└── NO → Are there shocks/discontinuities?
    ├── YES → Use upwind, TVD, or WENO
    └── NO → Use central with limiters
```

### Quick Reference

| Situation | Recommended Scheme |
|-----------|-------------------|
| Smooth, periodic | Central, spectral |
| Smooth, bounded | Central + one-sided BCs |
````
END THIRD-PARTY FILE f835a971bc0f7408

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