StencilDifferenzen

StencilDifferenzen, often referred to in English as stencil differences, denote a family of numerical techniques for approximating derivatives by forming linear combinations of function values at a fixed pattern of grid points, called a stencil, centered at the point where the derivative is to be estimated. The weights of the combination are chosen to match the derivative up to a desired order of accuracy and to respect the geometry of the grid.

In one dimension, a simple central first-derivative stencil uses three points: (u_{i+1} − u_{i−1})/(2h), yielding a second-order accurate approximation. The second-derivative central stencil is (u_{i+1} − 2u_i + u_{i−1})/h^2, also second-order accurate. Higher-order stencils employ more grid points, trading a wider stencil for higher-order error terms.

In two dimensions, the standard five-point stencil for the discrete Laplacian at (i,j) is (u_{i+1,j} + u_{i−1,j} + u_{i,j+1} + u_{i,j−1} − 4u_{i,j})/h^2, with O(h^2) error. Nine-point variants add diagonal neighbors to improve isotropy or accuracy. Stencil patterns extend similarly to three dimensions.

Stencils are linear operators and can be viewed as convolutions on regular grids or as sparse matrices in a finite-difference discretization. Near boundaries, one-sided stencils or ghost points are used, and on nonuniform grids the weights depend on local spacing.

StencilDifferenzen underpin finite difference methods for solving partial differential equations, including Poisson, heat, and wave equations, offering a simple and efficient approach on structured grids.