A Combination of High-Order Compact Finite Difference Schemes and a Splitting Method that Preserves Accuracy for the Multi-Dimensional Burgers' Equation

A class of high-order compact finite difference schemes combined with a splitting method that preserves accuracy are presented for numerical solutions of the multi-dimensional Burgers' equation. Firstly, the implicit high-order compact difference scheme is used to discretize Burgers' equation by non-linear weights that are required to be calculated at each time stage. Secondly, the sixth-order compact difference scheme in space and the fourth-order Runge-Kutta in time are applied to solve the 1D Burgers' equation. Meanwhile a linear stability analysis indicates the scheme is conditionally stable. Thirdly, the 2D and 3D Burgers' equations are divided into 1D subsystems by the splitting method, then these sub-equations' spatial terms are discretized by the fourth-order compact difference scheme, whereas the time discretizations are unchanged. The analyses of stability and accuracy of the splitting method are given to prove the accuracy of splitting without a significant loss. Finally, the accuracy and reliability of the proposed method are tested by comparing our experimental results with others selected from the available literature. It is shown that the new method has high-resolution properties and can effectively calculate Burgers' equation at large Reynolds number.