In a recent paper LeFloch and Thanh (2011) constructed the Riemann solutions for shallow water equations with discontinuous topography, where the stationary wave is determined by solving steady state equations. Here we consider the case when the original steady equations do not have a solution automatically. We thus construct the supplement Riemann solutions and prove that a backward shock wave emits to decelerate the flow which makes the steady equations solvable. Global entropy condition is imposed subsequently to select the physical relevant solution which states that the defined energy function should not only increase but increase to the maximum rate. Numerical results are given to verify our analysis.

在最近的一篇论文中，LeFloch和Thanh（2011）构造了具有不连续地形的浅水方程组的Riemann解，其中，稳态波是通过求解稳态方程组来确定的。在这里，我们考虑原始稳态方程没有自动解的情况。因此，我们构造了补充的Riemann解，并证明了向后激波的发出使流减速，从而使稳定方程可解。随后施加全局熵条件以选择物理相关解决方案，该解决方案指出已定义的能量函数不仅应增加，而且应增加到最大速率。数值结果证明了我们的分析。