In this study, the space–time (ST) generalized finite difference method (GFDM) was combined with Newton’s method to stably and accurately solve two-dimensional unsteady Burgers’ equations. In the coupled ST approach, the time axis is selected as a spatial axis; thus, the temporal derivative in governing equations is treated as a spatial derivative. In general, the GFDM is an optimal meshless collocation method for solving partial differential equations. Moreover, one can avoid the construction of a mesh for simulation by using the GFDM. The derivatives at each node are described as a linear combination of nearby functional values by using weighting coefficients in the computational domain. Due to the property of the moving least-square approximation in the GFDM, the resultant matrix system can be formed as a sparse matrix so that the GFDM is suitable for solving large-scale problems. In this study, two benchmark examples were used to demonstrate the consistency and accuracy of the proposed ST meshless numerical scheme.

在这项研究中，时空（ST）广义有限差分法（GFDM）与牛顿法相结合，可以稳定，准确地求解二维非定常Burgers方程。在耦合ST方法中，时间轴被选为空间轴；因此，控制方程中的时间导数被视为空间导数。通常，GFDM是求解偏微分方程的最佳无网格搭配方法。而且，可以避免使用GFDM构建用于仿真的网格。通过在计算域中使用加权系数，将每个节点的导数描述为附近功能值的线性组合。由于GFDM中移动最小二乘近似的性质，所得矩阵系统可以形成为稀疏矩阵，因此GFDM适合解决大规模问题。在这项研究中，使用两个基准示例来证明所提出的ST无网格数值格式的一致性和准确性。