浅水动边界问题的位移法模拟

为精确模拟浅水波非线性演化过程中的动边界，提出一种基于位移的Hamilton变分原理，并进而导出一种基于位移的浅水方程（Shallow Water Equation based on Displacement，SWE D）.SWE D以位移为基本未知量，可以精确满足动边界处的零水深要求并精确捕捉动态边界位置，且解具有协调性.在Hamilton变分原理的框架下，分别采用有限元和保辛积分算法对该浅水方程进行空间离散和时间积分，可有效地处理不平水底情况，保证对非线性演化进行长时间仿真的精度.数值算例表明该方法适用于浅水动边界问题的数值模拟.

Simulation on moving boundaries of shallow water using displacement method

To exactly simulate the moving boundaries of shallow water flow in the procedure of nonlinear evolution, a Hamilton variational principle based on displacement is proposed. Furthermore, a Shallow Water Equation based on Displacement (SWE D) is developed in terms of the Hamilton variational principle. Taking the displacement as a basic unknown variable, SWE D can exactly satisfy the requirement of zero water depth at moving boundary and exactly capture the location of the moving boundary, and the solutions are well balanced. In the frame of Hamilton variational principle, the finite element method and symplectic method are respectively used for the spatical discretization and time integral, which can effectively deal with the uneven water bottom and keep the accuracy in simulating the long time nonlinear evolution. The numerical examples show that the method is suitable to the simulation on the shallow water flow with moving boundaries.