Fast explicit operator splitting method for convection–diffusion equations

Abstract

AbstractSystems of convection–diffusion equations model a variety of physical phenomena which often occur in real life. Computing the solutions of these systems, especially in the convection dominated case, is an important and challenging problem that requires development of fast, reliable and accurate numerical methods. In this paper, we propose a second‐order fast explicit operator splitting (FEOS) method based on the Strang splitting. The main idea of the method is to solve the parabolic problem via a discretization of the formula for the exact solution of the heat equation, which is realized using a conservative and accurate quadrature formula. The hyperbolic problem is solved by a second‐order finite‐volume Godunov‐type scheme.We provide a theoretical estimate for the convergence rate in the case of one‐dimensional systems of linear convection–diffusion equations with smooth initial data. Numerical convergence studies are performed for one‐dimensional nonlinear problems as well as for linear convection–diffusion equations with both smooth and nonsmooth initial data. We finally apply the FEOS method to the one‐ and two‐dimensional systems of convection–diffusion equations which model the polymer flooding process in enhanced oil recovery. Our results show that the FEOS method is capable to achieve a remarkable resolution and accuracy in a very efficient manner, that is, when only few splitting steps are performed. Copyright © 2006 John Wiley & Sons, Ltd.