Exponential higher-order compact scheme for 3D steady convection–diffusion problem

Abstract

In this paper, the exponential higher order compact (EHOC) finite difference schemes proposed by Tian and Dai (2007) [10] for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case. The proposed EHOC scheme has the feature that it provides very accurate solution (exact in case of constant convection coefficient in 1D) of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil. The key properties of this scheme are its stability, accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes. To validate the present EHOC method, three test problems, mostly with boundary or internal layers are solved. Comparisons are made between numerical results for the present EHOC scheme and other available methods in the literature.