differentiation-schemes

Finite-difference stencils are numerical approximations of derivatives on discrete grids. Use the stencil generator to specify derivative order, desired accuracy order, scheme type (central, forward, backward), and grid spacing to produce stencil offsets and coefficients optimized for your discretization.

Scheme selection depends on field smoothness, boundary conditions, and accuracy requirements. This Skill recommends appropriate schemes by analyzing your grid type, boundary treatment needs, and smoothness characteristics, then outputs truncation-error estimates and dispersion/dissipation analysis to guide your choice.

Boundaries require specialized stencils since standard interior schemes cannot extend beyond the domain. Specify your boundary type during stencil generation; the Skill constructs one-sided or modified stencils at edges while maintaining accuracy and estimating truncation error near boundaries.

Truncation-error estimation quantifies the discretization accuracy loss. The Skill computes error bounds for your selected scheme, analyzes dispersion and dissipation across smooth and non-smooth fields, and outputs validation data to assess stability implications before deploying your discretization.

Differentiation-scheme generation requires Python 3.8 or later and NumPy; no additional heavy dependencies are needed. The Skill outputs results via scripted JSON, making integration straightforward in any Python-based simulation or discretization workflow.

Higher accuracy orders reduce truncation error but require wider stencils and smoother input data. Specify your target accuracy order and field smoothness to the generator; it constructs wider stencils with optimized coefficients and returns error metrics showing accuracy gain versus computational cost.