Title: Two-stage explicit schemes based on numerical approximations of convection-diffusion equations

Authors: Mohammad Izadi

Addresses: Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman-7616913439, Iran

Abstract: A wide family of finite difference methods for the convection-diffusion problems based on an explicit two-stage scheme and a seemingly implicit method are presented. In this paper, to have a greater stability region while maintaining the second-order accuracy, a family of methods that combines the MacCormack and Saul'vey schemes is proposed. The stability analysis of the combined methods is investigated using the von Neumann approach. In each case, it is found that it is the convection term that limits the stability of the scheme. Based on the von Neumann analysis, valuable stability limits in terms of mesh parameters for maintaining accurate results are determined in an analytic manner and demonstrated through computer simulations. Two model problems consisting of linear advection-diffusion and nonlinear viscous Burgers equation are given to illustrate some properties of the present technique, such as stability and the ability to propagate discontinuities.

Keywords: advection-diffusion; Burgers equation; convection-diffusion equations; finite difference approximation; MacCormack scheme; Saul'yev scheme; stability analysis.

DOI: 10.1504/IJCSM.2022.128185

International Journal of Computing Science and Mathematics, 2022 Vol.16 No.3, pp.208 - 224

Accepted: 24 Jul 2020
Published online: 11 Jan 2023 *

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