Higher-Order Exponential Difference Schemes for the Computations of the Steady Convection–Diffusion Equation

Abstract

Conventional exponential difference schemes may yield accurate and stable solutions for the one-dimensional, source-free convection–diffusion equation. However, its accuracy will be deteriorated in the presence of a nonconstant source term or in multidimensional problems. Attempts are made to increase the accuracy of exponential difference schemes. First, we propose an exponential difference scheme that retains second-order accuracy in the presence of a source term or in multidimensional situations. Mathematical analysis and numerical experiments are performed to validate this scheme. Second, a local particular solution method is introduced to raise the solution accuracy for problems with a source term. This method locally transforms the original problem to a source-free one, to which an accurate solution can be obtained. Performance of this process is verified by numerical calculations of some test problems. Third, two skew exponential difference schemes are proposed to raise the solution accuracy in multidimensional problems: one is designed to be free of numerical diffusion and the other with minimum numerical diffusion to ensure solution monotonicity. Comparisons with existing schemes are performed by conducting numerical experiments on several test problems. Finally, a simple blending procedure of these two schemes is suggested to yield an accurate and stable representation of the convection–diffusion problem in all possible situations, with or without solution discontinuities.