A modified central discontinuous Galerkin method with positivity-preserving and well-balanced properties for the one-dimensional nonlinear shallow water equations

In this paper, we present a modified central discontinuous Galerkin method for solving the nonlinear shallow water equations over variable bottom topography in one-dimensional space. The proposed method balances exactly the flux gradient and source term in the still-water stationary case by adding a correction term to the base scheme and ensures non-negativity of the water depth by using special approximations to the bottom together with a positivity-preserving limiter, meanwhile reduces the computational cost considerably compared with the standard central discontinuous Galerkin method by using L2 projection to obtain an approximation solution. Numerical tests are presented to illustrate the accuracy and validity of the proposed schemes.