Asymptotically Stable Fourth-Order Accurate Schemes for the Diffusion Equation on Complex Shapes

Abstract

An algorithm which solves the multidimensional diffusion equation on complex shapes to fourth-order accuracy and is asymptotically stable in time is presented. Thisbounded-errorresult is achieved by constructing, on arectangular grid,a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty-like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions, fail. The ability of the paradigm to be applied to arbitrary geometric domains is an important feature of the algorithm.