固定网格下的特征线法求解溃坝问题

Characteristic line method under fixed meshes for the simulation of dambreak

溃坝问题是典型的非线性双曲方程的Riemann问题,其数值求解的难点在于对间断面的捕捉以及避免间断面处在数值计算过程中产生数值色散,因而为求解此问题所产生的各种数值计算方法的优劣也体现在这两个方面。本文针对溃坝问题提出一种新的计算方法。该方法基于对偶变量推导的浅水波方程,根据方程的特点,从方程的特征值和黎曼不变量出发,采用高精度的激波捕捉方法计算黎曼不变量的位置随时间的变化,然后映射至不随时间变化的固定网格。根据黎曼不变量的位置,采用保形分段三次Hermite插值将物理量映射至网格节点。计算结果显示,该方法不仅操作简单,计算量小,而且结果准确。

Dambreak is a typical Riemann problem of nonlinear hyperbolic equations.The difficulty of solving a Riemann problem is the difficulty in capturing the discontinuous interface,at which numerical dispersion appears.Therefore,there are the two key aspects in evaluating the numerical methods for solving a Riemann problem.In this paper,a new computational method is proposed for the calculation of dambreak.Nonlinear equations for shallow water waves are established based on dual variables.The computational scheme is based on the eigenvalues and Riemann invariants of the equations,and focuses on capturing the location of Riemann invariants with high-precision shock capturing method before they are mapped to the fixing meshes by the interpolation function of Piecewise Cubic Hermite Interpolating Polynomial.Numerical results verify that the method is accurate and efficient,at the same time the numerical procedure is easily operated.