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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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Atualizado18 de maio de 2026 às 08:58

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Perform spatial and temporal convergence analysis for solution verification — compute observed convergence orders from grid or timestep refinement studies, apply Richardson extrapolation to estimate discretization error, and calculate the Grid Convergence Index (GCI) per ASME V&V 20 standards. Use when verifying that a numerical solution converges at the expected rate, estimating the error on the finest mesh, checking whether grids are in the asymptotic range, or preparing formal verification reports, even if the user only asks "is my mesh fine enough" or "how accurate is my solution."

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linear-solvers

HeshamFS/materials-simulation-skills

Select and configure linear solvers for Ax=b systems arising in numerical simulations — choose between direct (LU, Cholesky) and iterative (CG, GMRES, BiCGSTAB, MINRES) methods, analyze sparsity patterns and matrix conditioning, recommend preconditioners (AMG, ILU, IC), apply row/column scaling, and diagnose convergence stagnation from residual histories. Use when setting up a linear solve for FEM/FVM assembly, debugging slow or stalled Krylov iterations, choosing a preconditioner for SPD or nonsymmetric systems, or investigating ill-conditioning, even if the user only says "my solver is slow" or "GMRES won't converge."

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mesh-generation

HeshamFS/materials-simulation-skills

Plan and evaluate mesh generation for numerical simulations — estimate grid resolution from physics scales (interface width, boundary layers, wavelengths), check aspect ratios and skewness against quality thresholds, choose between structured, unstructured, and adaptive mesh refinement strategies, and compute grid sizing for 1D/2D/3D domains. Use when setting up a new mesh, diagnosing poor solver convergence caused by mesh quality, deciding how many points to place across a phase-field interface or boundary layer, or preparing a mesh convergence study, even if the user only asks "what resolution do I need" or "why is my solver failing."

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nonlinear-solvers

HeshamFS/materials-simulation-skills

Select and configure nonlinear solvers for root-finding f(x)=0, optimization min F(x), and least-squares problems — choose among Newton, Newton-Krylov, quasi-Newton (BFGS, L-BFGS), Broyden, Anderson acceleration, and Levenberg-Marquardt methods, configure line search or trust-region globalization, diagnose convergence rate (quadratic, linear, stagnated), and assess Jacobian quality and conditioning. Use when a Newton solver converges slowly or diverges, choosing between line search and trust region, debugging nonlinear iteration failures in FEM or phase-field codes, or selecting a solver for large-scale unconstrained optimization, even if the user only says "my Newton iterations aren't converging."

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numerical-integration

HeshamFS/materials-simulation-skills

Select and configure time integration methods for ODE and PDE simulations — choose among explicit Runge-Kutta, BDF, Rosenbrock, and Adams families, set relative and absolute error tolerances, implement adaptive step-size control with I/PI/PID controllers, plan IMEX operator splitting for mixed stiff and non-stiff terms, and estimate splitting errors. Use when picking an integrator for a new simulation, diagnosing step rejections or tolerance failures, setting up operator splitting for phase-field or reaction-diffusion problems, or deciding between explicit and implicit time marching, even if the user only says "my solver keeps rejecting steps" or "which ODE method should I use."

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numerical-stability

HeshamFS/materials-simulation-skills

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."

2026-05-1840

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HeshamFS

HeshamFS/materials-simulation-skills

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EstatísticosInformática e Matemática15-2041L4

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