A semi-Lagrangian Bernstein–Bézier finite element method for two-dimensional coupled Burgers’ equations at high Reynolds numbers

Abstract

This paper aims to develop a semi-Lagrangian Bernstein–Bézier high-order finite element method for solving the two-dimensional nonlinear coupled Burgers’ equations at high Reynolds numbers. The proposed method combines the semi-Lagrangian scheme for the time integration and the high-order Bernstein–Bézier functions for the space discretization in the finite element framework. Unstructured triangular Bernstein–Bézier patches are reconstructed in a simple and inherent manner over finite elements along the characteristic curves defined by the material derivative. A fourth-order Runge–Kutta scheme is used for the approximation of departure points along with a local L2-projection to compute the solution at the semi-Lagrangian stage. By using these techniques, the nonlinear problem is decoupled and two linear diffusion problems are solved separately for each velocity component. An implicit time-steeping scheme is used and a preconditioned conjugate gradient solver is used for the resulting linear systems of algebraic equations. The proposed method is investigated through several numerical examples including convergence studies. It is found that the proposed method is stable, highly accurate and efficient in solving two-dimensional coupled Burgers’ equations at high Reynolds numbers.