Differentiation Schemes

Goal

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

Requirements

• Python 3.8+

• 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]

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