基于无结构化网格浅水方程的隐式解法

An implicit scheme with unstructured grids for shallow water equation

为提高有限体积法计算浅水方程的数值稳定性,采用Roe方法近似Riemann解计算界面通量,利用TVD-MUSCL格式对守恒变量进行重构,推导并建立了高效的隐式计算格式。该格式基于无结构化网格,将计算精度提高到二阶,使用面积加权计算流速梯度,对坡底梯度的处理满足静水平衡要求。为在时间积分上采用隐格式,又导出了控制方程的雅可比矩阵全解析形式,并采用牛顿-拉夫逊方法进行迭代求解。通过对溃坝算例的对比分析,验证该方法的稳定性、和谐性和高效性,同时也证明该算法具有捕捉溃坝激波问题的能力。

In order to improve the numerical stability of shallow water equation calculated by Finite Volume Method, by emplo- ying Roe＇s approximate Riemann solution to calculate the interface flux and TVD - MUSCL Format to reconstruct the conservation variable, a high efficient implicit computation scheme is derived. On the basis of the unstructured grids, this format improves the computation accuracy to grade 2. It computes the velocity gradient by the area weight and satisfies the stationary hydraulic pres- sure equilibrium by handling the bed slope term. In order to use the implicit scheme for the time integration, the full resolution form of Jacobian matrix is analytically derived, which was solved by Newton - Raphson algorithm iteratively. By the comparison with various numerical studies on dam - breaking cases, this computation method is proved to be stable, compatible and efficient with the capability of accurately capturing the shock wave in dam - breaking problems.