Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes
A finite difference scheme for a multidimensional diffusion equation with constant coefficients in a domain with complex shape is described. The scheme in based on a rectangular grid, has fourth-order spatial approximation, and is asymptotically stable in time. The error estimate is based on the property that the symmetric part of the matrix corresponding to the spatial discretization is negative definite. The Dirichlet boundary conditions are imposed in a penalty-like manner. Several numerical tests for two-dimensional problems show that the method is stable, accurate, and robust. This paper is an interesting example of how the traditional differencing technique combined with a good mathematical insight can lead to efficient schemes for domains with complex shape.
scientific article; zbMATH DE number 1036507