Higher-Order Finite-Difference Methods for the Burgers’ Equations

We propose several higher-order explicit finite difference methods (FDMs) for solving one- and two-dimensional Burgers’ equations, as well as two-dimensional coupled Burgers’ equations with a corresponding initial condition and boundary conditions. The proposed FDMs for solving one- and two-dimensional Burgers’ equations have a sixth-order approximation in space variables and a third-order approximation in the time variable, while for solving two-dimensional coupled Burgers’ equations, they have a fourth-order approximation in space variables and a second-order approximation in the time variable. A numerical method for solving one-dimensional Burgers’ equation using the relationship between the heat equation and Burgers’ equation is also presented. This method has a sixth-order approximation in the space variable. The accuracy of the proposed method is demonstrated by some test problems. The numerical results are in good agreement with the exact solutions.