中文摘要：

研究二维浅水波方程的数值激波不稳定性问题．线性稳定性分析和数值实验表明，格式的临界稳定性与数值激波的不稳定现象有重要的联系．基于扰动量的增长矩阵分析，本文将高分辨率的数值格式和HLL格式进行特定的加权，设计一类新的混合型数值格式．其中可以调节非线性波速的HLLC与HLL的混合格式，数值试验展示了消除浅水波方程激波不稳定现象的有效性和鲁棒性．

英文摘要：

In calculation of multidimensional fluid mechanics problems with numerical schemes that accurately capture contact discontinuity, perturbation near shock wave may increase dramatically. This is called numerical shock instability. In this paper numerical shock instability on shallow water equations is studied. By analyzing linear stability of several numerical schemes, marginal stability of schemes are found having close relation with numerical shock instability. According to eigenvalue analysis, a hybrid method is designed to remedy nonphysical phenomenon by locally modify the original schemes. Numerical experiments show efficiency and robustness of HLLC-HLL hybrid scheme in eliminating shock instability of shallow water equations.