A moving finite element method for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers

Abstract

The coupled Burgers’ equations at high Reynolds numbers usually have sharp gradients or are discontinuous in the solution. Therefore, it is difficult to obtain analytical solutions. This paper aims to use the moving finite element method proposed by Li et al. (2001) to get stable and high-precision numerical solutions for the coupled Burgers’ equations at high Reynolds numbers. The method decouples the mesh equation and partial differential equation (PDE) into two unrelated parts, mesh reconstruction and PDE solver. The mesh reconstruction constructs the harmonic mapping between the physical and logical domains through an iterative method so that the mesh structure maintains harmonics after multiple numerical integrations. We compute three classic numerical examples. Numerical results show that the moving finite element method effectively solve the coupled Burgers’ equations at high Reynolds numbers, obtain stable numerical results, and achieve higher numerical accuracy. During the evolution of the solution, the mesh is always concentrated in the position where the solution has sharp gradients.