Efficient and accurate numerical methods for the multidimensional convection–diffusion equations

Abstract

A class of high order compact methods combined with local one-dimensional method have been studied to numerically solve multidimensional convection–diffusion equations. The methods are widely accepted due to their compactness, high accuracy. In this kind of methods the spatial derivatives are approximated implicitly rather than explicitly with smaller stencil but with higher accuracy. The local one-dimensional strategy is adopted in time to reduce the scale of algebraic equations resulting from numerical methods. This makes the multidimensional problems be easily coded. Based on analyzing the splitting error of the local one-dimensional method, a more accurate scheme is obtained through minor modification on the original scheme. By Von Neumann approach, we can find that the proposed schemes are unconditionally stable. Some numerical results are reported to illustrate that the schemes are robust, efficient and accurate.