A semi-Lagrangian approach for numerical simulation of coupled Burgers' equations

In this study, we develop a numerical method for solving the coupled viscous Burgers' equations based on a backward semi-Lagrangian method. The main difficulty associated with the backward semi-Lagrangian method for this problem is treating the nonlinearity in the diffusion-reaction equation, whose reaction coefficients are given in terms of coupled partial derivatives. To handle this difficulty, we use an extrapolation technique which splits the nonlinearity into two linear diffusion-reaction boundary value problems. In the proposed backward semi-Lagrangian method, we use fourth-order finite differences to discretize the diffusion-reaction boundary value problems and employ the so-called error correction method to solve the highly nonlinear initial value problems. Our overall algorithm is completely iteration-free and computationally efficient. We demonstrate the numerical accuracy and efficiency of the present method by comparing our numerical results with analytical solutions and other numerical solutions based on alternative existing methods.