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 |
