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Selects numerical differentiation schemes (finite-difference, compact, spectral, WENO), generates stencils at arbitrary order, handles boundary stencils, and estimates truncation error for PDE/ODE discretization.
How this skill is triggered — by the user, by Claude, or both
Slash command
/full:differentiation-schemesThis skill is limited to the following tools:
The summary Claude sees in its skill listing — used to decide when to auto-load this skill
Provide a reliable workflow to select a differentiation scheme, generate stencils, and assess accuracy for simulation discretization.
Provide a reliable workflow to select a differentiation scheme, generate stencils, and assess accuracy for simulation discretization.
| 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 |
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
| 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 | Key Outputs |
|---|---|
scripts/stencil_generator.py | offsets, coefficients, order, accuracy, scheme |
scripts/scheme_selector.py | recommended, alternatives, notes |
scripts/truncation_error.py | error_scale, order, reduction_if_halved |
scripts/scheme_selector.pyscripts/stencil_generator.pyscripts/truncation_error.pyUser: I need to discretize a second derivative for a diffusion equation on a uniform grid. I want 4th-order accuracy.
Agent workflow:
--boundary to surface one-sided/ghost-cell
guidance, or ask the user):
python3 scripts/scheme_selector.py --smooth --order 2 --accuracy 4 --json
python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json
[-1/12, 4/3, -5/2, 4/3, -1/12] / dx².# 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 --accuracy 2 --scale 1.0 --json
| Error | Cause | Resolution |
|---|---|---|
order must be positive | Invalid derivative order | Use 1, 2, 3, ... (max 6) |
order must be <= 6 | Derivative order too large | Use 1–6 |
accuracy must be even for central | Odd accuracy requested for central scheme | Use 2, 4, 6, ... |
scheme must be central, forward, or backward | Invalid --scheme value | Use central, forward, or backward |
| Property | Meaning |
|---|---|
| Symmetric offsets | Central scheme (no directional bias) |
| Asymmetric offsets | One-sided or upwind scheme |
| More points | Higher accuracy but wider stencil |
| 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 |
| 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] |
Before trusting a generated stencil or accepting a scheme recommendation, record concrete evidence for each item below:
stencil_generator.py --json and confirmed results.accuracy matches the requested order AND that len(results.offsets) equals the expected stencil width (e.g. 5 points for a 4th-order central second derivative); for a central scheme also verified the offsets are symmetric about 0.coefficients against references/stencil_catalog.md: confirmed they sum to ~0 (consistency: the operator annihilates a constant) and reproduce a known catalog stencil for at least one standard case (e.g. 2nd-order d²/dx² gives [1, -2, 1]/dx²).central scheme, confirmed --accuracy is even (odd values exit 2 with accuracy must be even for central); recorded the actual exit code rather than assuming the requested order was achieved.error_scale, order, and reduction_if_halved from truncation_error.py and confirmed reduction_if_halved == 2**accuracy, then compared error_scale against the smallest physical feature size (dx/L_feature from references/error_guidance.md) to confirm the grid actually resolves the physics.p_obs = log(e_h/e_{h/2})/log(2) is within ~10% of the formal accuracy before quoting that order.--scheme forward|backward or --boundary guidance) per references/boundary_handling.md, since the interior stencil alone does not define the scheme order at the boundary.scheme_selector.py did NOT recommend high-order central FD and that a limiter/WENO/upwind path was chosen instead.| Tempting shortcut | Why it's wrong / what to do |
|---|---|
| "The stencil generator returned coefficients, so the scheme is the order I asked for." | The accuracy field just echoes your request; it is not measured. Verify the achieved order with a grid-refinement study and confirm the coefficients match a catalog stencil and sum to ~0. |
"I asked for 4th order on a central scheme with --accuracy 3, it'll just round up." | It will not — central schemes reject odd accuracy with accuracy must be even for central (exit 2). Pass an even accuracy; an odd request is an error, not a silent upgrade. |
| "Higher accuracy order always means lower error here." | truncation_error.py reports asymptotic scaling (scale * dx**accuracy); for a coarse grid or under-resolved feature the higher-order term need not dominate, and roundoff (O(ε/dx^p)) can win on very fine grids. Compare error_scale to the feature size, do not assume monotone improvement. |
| "Two grids agree closely, so it's converged." | Two grids cannot estimate observed order or confirm the asymptotic range. Use >=3 grids and compute p_obs before claiming the formal order (see references/error_guidance.md). |
| "The interior stencil is 4th order, so my whole solve is 4th order." | The generator emits interior stencils only; boundary closures often limit the global order. Generate one-sided/ghost-cell stencils explicitly and verify the boundary does not drop the observed order (references/boundary_handling.md). |
| "The field has a shock but a wide central stencil is more accurate, so use it." | High-order central FD oscillates (Gibbs) at discontinuities. scheme_selector.py recommends FV with limiter/WENO or upwind for --discontinuous; follow it rather than maximizing formal order. |
"Custom --offsets let me build any stencil I want." | Offsets must be distinct, length-capped (51), and number more than the derivative order, or the script exits 2. A valid run still does not guarantee the intended accuracy — verify the coefficients and observed order. |
--order (derivative order) is validated as a positive integer with an upper bound (order <= 6)--accuracy is validated as a positive integer (<= 8), and additionally must be even for central schemes--scheme is validated against a fixed allowlist (central, forward, backward)--offsets (custom stencil) is length-capped (max 51), parsed as distinct integers, and must exceed the derivative order--dx and --scale are validated as finite, non-negative numbers (--dx strictly positive)stencil_generator.py, scheme_selector.py, truncation_error.py) with explicit argument listseval(), exec(), or dynamic code generationshell=True)references/stencil_catalog.md - Common stencilsreferences/boundary_handling.md - One-sided schemesreferences/scheme_selection.md - FD/FV/spectral comparisonreferences/error_guidance.md - Truncation error scalingnpx claudepluginhub heshamfs/materials-simulation-skills --plugin core-numericalChecks CFL and Fourier stability criteria for PDE simulations, performs von Neumann analysis, detects stiffness, and recommends explicit vs implicit time-stepping.
Classifies ODE/PDE type, selects analytical or numerical solution methods, validates solutions, and interprets physical meaning.
Analyzes diffusion processes via SDEs, Fokker-Planck equations, first-passage times, and Monte Carlo validation. Use for probability density evolution, mean first-passage times, or parameter sensitivity analysis.