Numerical solution of elliptic partial differential equations in discontinuous domains using a higher order accurate finite difference scheme

Abstract

In this paper, we employ a higher order accurate finite difference scheme [1] for numerically solving an elliptic equation with discontinuous coefficients and singular source terms. Here we consider the most general form of elliptic equation which admits discontinuities of the dependent variable, the flux, the convection coefficient and the source term. This new scheme is obtained by coupling a recently developed Higher Order Compact (HOC) methodology with special treatment for the points just next to the points of discontinuity. The scheme is proficient in handling jumps across the interface quite efficiently and the overall order of accuracy of the scheme is at least two. The scheme is used to solve two-dimensional (2D) problems in polar coordinates on a non-uniform space grid. The grid is constructed in such a way that the grid points cluster around the circular interface (points of discontinuity), where the points of discontinuity themselves are not nodes. By wisely choosing the intensity of clustering around the points of discontinuity, which can be easily implemented in actual programming, one can minimize the overall error in computing. Numerous numerical studies on a number of problems are conducted and results are compared with those obtained with immersed interface and other well-known methods in the literature. The numerical results demonstrate that the proposed method provides outstanding results on relatively coarser grids.