Saint-Venant方程组全隐式标量耗散有限体积法

摘要

基于Saint-Venant方程组的守恒形式,重构了各物理变量在单元格边界的黎曼状态值,实现了各变量在计算区域内的二阶精度分布.在此基础上,构造了对流通量项的具有标量耗散特征的有限体积法,并在地表水位相对高程梯度离散式中引入额外空间离散项,该项在有水区域为零,并在无水区域能与地表水位相对高程梯度项相互抵消,从而正确描述地表水位相对高程梯度的真实作用.采用双时间步法对Saint-Venant方程组的空间离散式进行全隐式离散,实现了无条件稳定求解.选取了2个典型算例,采用数量呈倍数递减的3种时间步长进行数值模拟,通过与解析解和实测结果进行对比,验证了数值解法的模拟效果和收敛性.结果表明,建立的数值解法能以优良的拟合度模拟不同断面几何约束下的溃坝过程,模拟结果表现出了良好的收敛性.%Based on the conservative form of Saint-Venant equations,the Riemann state value of each physical variable in the cell boundary was reconstructed,and the two order accuracy distribution of each variable in the computational domain was realized.On this basis,the finite-volume scheme with scalar dissipation is constructed for the convective fluxes.In order to describe the real physical impact of the relative elevation gradient of water level,extra space discretization is added in the discrete term of relative elevation gradient of water level.This term is equal to zero at the wetted region,and can be counteracted by the term of relative elevation gradient of water level at the dry area.Dural time-step algorithm is used for the fully implicit discretization the time terms in Saint-Venant equations.Therefore,the unconditional stability of solution is achieved.Finally,two typical examples are given to verify the stability and convergence of this new numerical solution.