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differentiation-schemes

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."

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npx skills add https://github.com/HeshamFS/materials-simulation-skills --skill differentiation-schemes

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HeshamFS/materials-simulation-skills

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UpdatedMay 18, 2026 at 08:58

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HeshamFS/materials-simulation-skills

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HeshamFS/materials-simulation-skills

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HeshamFS/materials-simulation-skills

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Source

HeshamFS

HeshamFS/materials-simulation-skills

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StatisticiansComputer and Mathematical Occupations15-2041L4

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	{"author":"HeshamFS","version":"1.1.0","security_tier":"high","security_reviewed":true,"tested_with":["claude-code","gemini-cli","vs-code-copilot"],"eval_cases":4,"last_reviewed":"2026-03-26"}

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
Advection-dominated	Upwind
Shocks/fronts	TVD, WENO
High accuracy needed	Compact (Padé), spectral

Script Outputs (JSON Fields)

Script	Key Outputs
`scripts/stencil_generator.py`	`offsets`, `coefficients`, `order`, `accuracy`
`scripts/scheme_selector.py`	`recommended`, `alternatives`, `notes`
`scripts/truncation_error.py`	`error_scale`, `order`, `notes`

Workflow

Identify requirements - derivative order, accuracy, smoothness
Select scheme - Run scripts/scheme_selector.py
Generate stencils - Run scripts/stencil_generator.py
Estimate error - Run scripts/truncation_error.py
Validate - Test with manufactured solutions or grid refinement

Conversational Workflow Example

User: I need to discretize a second derivative for a diffusion equation on a uniform grid. I want 4th-order accuracy.

Agent workflow:

Select appropriate scheme:

python3 scripts/scheme_selector.py --smooth --periodic --order 2 --accuracy 4 --json

Generate the stencil:

python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json

Result: 5-point stencil with coefficients [-1/12, 4/3, -5/2, 4/3, -1/12] / dx².

Pre-Discretization Checklist

Confirm derivative order and target accuracy
Choose scheme appropriate to smoothness and boundaries
Generate and inspect stencils at boundaries
Estimate truncation error vs physics scales
Verify with grid refinement study

CLI Examples

# Select scheme for smooth periodic problem
python3 scripts/scheme_selector.py --smooth --periodic --order 1 --accuracy 4 --json

# Generate central difference stencil for first derivative
python3 scripts/stencil_generator.py --order 1 --accuracy 2 --scheme central --json

# Generate 4th-order second derivative stencil
python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json

# Estimate truncation error
python3 scripts/truncation_error.py --dx 0.01 --order 2 --accuracy 2 --scale 1.0 --json

Error Handling

Error	Cause	Resolution
`order must be positive`	Invalid derivative order	Use 1, 2, 3, ...
`accuracy must be even for central`	Odd accuracy requested	Use 2, 4, 6, ...
`Unknown scheme`	Invalid scheme type	Use central, upwind, compact

Interpretation Guidance

Stencil Properties

Property	Meaning
Symmetric offsets	Central scheme (no directional bias)
Asymmetric offsets	One-sided or upwind scheme
More points	Higher accuracy but wider stencil

Truncation Error Scaling

Accuracy Order	Error Scales As	Refinement Factor
2nd order	O(dx²)	2× refinement → 4× error reduction
4th order	O(dx⁴)	2× refinement → 16× error reduction
6th order	O(dx⁶)	2× refinement → 64× error reduction

Common Stencils

Derivative	Accuracy	Points	Coefficients (× 1/dx or 1/dx²)
1st	2	3	[-1/2, 0, 1/2]
1st	4	5	[1/12, -2/3, 0, 2/3, -1/12]
2nd	2	3	[1, -2, 1]
2nd	4	5	[-1/12, 4/3, -5/2, 4/3, -1/12]

Security

Input Validation

--order (derivative order) is validated as a positive integer with an upper bound
--accuracy is validated as a positive even integer for central schemes
--scheme is validated against a fixed allowlist (central, upwind, compact)
--dx and --scale are validated as finite positive numbers
No user-supplied strings are interpolated into code paths or shell commands

File Access

Scripts read no external files; all inputs are provided via CLI arguments
Scripts write only to stdout (JSON output); no files are created unless the agent explicitly uses the Write tool

Tool Restrictions

Read: Used to inspect script source, references, and user configuration files
Bash: Used to execute the three Python scripts (stencil_generator.py, scheme_selector.py, truncation_error.py) with explicit argument lists
Write: Used to save generated stencil coefficients or scheme recommendations; writes are scoped to the user's working directory
Grep/Glob: Used to locate relevant files and search references

Safety Measures

No eval(), exec(), or dynamic code generation
All subprocess calls use explicit argument lists (no shell=True)
Stencil computation uses only NumPy linear algebra on small, bounded matrices (stencil width limited by accuracy order)
All output is deterministic JSON with no shell-interpretable content

Limitations

Boundary handling: Stencil generator provides interior stencils; boundaries need special treatment
Nonuniform grids: Standard stencils assume uniform spacing
Spectral: Not covered by stencil generator

References

references/stencil_catalog.md - Common stencils
references/boundary_handling.md - One-sided schemes
references/scheme_selection.md - FD/FV/spectral comparison
references/error_guidance.md - Truncation error scaling

Version History

v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
v1.0.0: Initial release with 3 differentiation scripts