| Abstract: | In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer
shallow water equations, it is not a trivial work due to the complexity of the equations.
To achieve the well-balanced property of the numerical scheme easily, the two-layer
shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their
eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values
of the auxiliary variables suitably, we can prove that the scheme can exactly preserve
the still-water solution, and thus it is a truly well-balanced scheme. To ensure the
non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the
numerical method will be illustrated through some numerical tests. |
