An economical robust algorithm for solving 1D coupled Burgers’ equations in a semi-Lagrangian framework

Abstract

In this study, we propose an efficient algorithm for solving one-dimensional coupled viscous Burgers’ equations. One of the main accomplishments of this study is to develop a stable high-order algorithm for the system of reaction–diffusion equations. The algorithm is “robust” because it is designed to prevent non-physical oscillations through an iteration procedure of a block Gauss-Seidel type. The other is to develop an efficient algorithm for the Cauchy problem. For this, we first find half of the upstream points by adopting a multi-step qth-order (q=2,3) error correction method. The algorithm is also “economical” in the sense that an interpolation strategy for finding the remaining upstream points is designed to dramatically reduce the high computational cost for solving the nonlinear Cauchy problem without damage to the order of accuracy. Three benchmark problems are simulated to investigate the accuracy and the superiority of the proposed method. It turns out that the proposed method numerically has the qth-order temporal and 4th-order spatial accuracies. In addition, the numerical experiments show that the proposed method is superior to the compared methods in the sense of the computational cost.