浅水方程组的数值方法

A Numerical Method of Shallow Water Equations

构造了浅水方程组的二阶精度的TVD格式。格式由简单的TVD Runge-Kutta型时间离散和有坡度限制的空间对称离散格式组成。数值耗散项用局部棱柱化河道流的特征变量构造。格式的主要优点是能够计算天然河道中浅水方程组的弱解并且构造简单。格式能够求出天然河道或非平底部渠道中的精确静水解。给出了渠道溃坝问题数值解与解析解的比较,验证格式精度高。实际天然河道型梯级水库溃坝的数值实验表明格式稳定,适应性强。

A Second-order accurate TVD scheme for shallow water equations is presented. A simple TVD Runge-Kutta type time discretization and a symmetric space discretization based on the slope limiter is used. The mumerical dissipation term is constructed based on the local prism river flows. The main advantage of the scheme stems from the ability to compute the weak solutions of unsteady free-surface flows in the natural river courses and the simplicity. Constant still water solution in the natural river courses of the uneven bottom channel can be exactly compted. Verification of the scheme for the dam-break problem in the channel is made by comparison with analytical solution and good agreement is found. Numerical experiments with the dam-break problem in the practical natural cascade reservoirs show the scheme is stable and robust.