用最小二乘无网格有限差分方法求解二维浅水方程

Meshfree Least-Squares-Based Finite Difference Method for 2-D-Shallow Water Equation

近年来兴起的无网格方法摆脱了网格约束,在计算区域内能自由布点,快速灵活,并且离散节点能更好地拟合复杂的边界,求解精度较高。运用最小二乘无网格有限差分(MLSFD)方法,求解二维非守恒形式浅水方程。支撑域内点采用2种方法结合的方式进行选取。为避开繁琐的矩阵奇异、病态的检验过程,选用了最小二乘技术对矢量做最优化近似,并使用MLSFD对圆形溃坝模型进行计算。为了更好地处理边界条件,在无网格的四周,向内分别布置三层直角网格。对数值模拟结果进行分析比较,验证了该方法在计算二维浅水流动的数值模拟方面有着较高的精确度,进而说明无网格方法运用于求解浅水方程是可行的。

Meshfree methods could be used to deliberately distribute nodes freely and simply in the computational domain free from constraints. Compared with traditional methods, meshfree methods have greater ability to deal with complex boundary. In this paper, a Meshfree Least-Squares-based Finite Difference （MLSFD） method is applied to the calculation of 2-D non-conservative shallow water equations. Two solutions are presented for nodes selection. In order to avoid cumbersome examination process, the least square technique is introduced to optimize the approxima- tion of the vector. The MLSFD method is employed to simulate a circular dam-break model. To treat boundary con- dition better, three-layer right-angle grids are distributed from the boundary to the inside around the meshfree nodes. Numerical simulation results are analyzed and compared. The results show that the MLSFD method has a high accuracy of simulating 2-D shallow water flow. It＇ s indicated that the MLSFD method is feasible for solving shallow water equations.