In this paper, a class of two-level high-order compact finite difference implicit schemes are proposed for solving the Burgers’ equations. Firstly, based on the fourth-order compact finite difference schemes for spatial derivatives and the truncation error remainder correction method for temporal derivative, the high-order compact (HOC) difference method is introduced for solving the one-dimensional (1D) Burgers’ equation. At the same time, the stability of the scheme is analyzed by using the Fourier analysis method. Because only three grid points are involved in each time level. The Thomas algorithm can be directly used to solve the tridiagonal linear system. Then, this method is extended to solve the two-dimensional (2D) and three-dimensional (3D) coupled Burgers’ equations. Finally, numerical experiments are conducted to verify the accuracy and the reliability of the present schemes.

提出了一类两级高阶紧致有限差分隐式格式，用于求解Burgers方程。首先，基于空间导数的四阶紧致有限差分方案和时间导数的截断误差余量校正方法，引入了高阶紧致（HOC）差分方法来求解一维（1D）Burgers方程。同时，通过傅里叶分析法对方案的稳定性进行了分析。因为每个时间级别仅涉及三个网格点。Thomas算法可以直接用于求解三对角线性系统。然后，将该方法扩展为求解二维（2D）和三维（3D）耦合的Burgers方程。最后，