Expspit

Expspit is a grid-generation technique used in numerical simulations to create nonuniform grids by exponentially spacing points along coordinates. It is commonly employed to increase resolution in regions with sharp gradients or boundary layers while using fewer points overall.

The core idea is to apply an exponential map to a uniformly spaced index to obtain physical coordinates. In one dimension, a typical mapping from index i in [0, N] to physical coordinate x in [a, b] is x(i) = a + (b − a) · (exp(β·i/N) − 1) / (exp(β) − 1). The parameter β controls clustering: β > 0 concentrates points near a, β < 0 near b, and β = 0 yields a uniform grid. For symmetric clustering or clustering in a target region, the map can be adapted to place points around midpoints or other locations. In multiple dimensions, the technique is extended via tensor-product maps or coordinate transformations to generate structured nonuniform grids.

Applications include finite difference and finite element methods for solving partial differential equations such as Poisson, fluid dynamics, and electromagnetic problems, particularly when boundary layers or localized features are present. Advantages of expspit include higher accuracy in critical regions with fewer grid points and straightforward implementation within standard solvers. Limitations involve choosing appropriate β values for a given problem, potential complications from nonuniform discretization in diffusion or advection terms, and the extra care required when extending to complex geometries or unstructured meshes.

Implementation typically occurs within mesh generators or solver pipelines, with higher-dimensional grids constructed through product maps or adaptive refinement guided by error indicators.

See also: nonuniform grid, coordinate transformation, mesh generation, finite difference method, finite element method.