A high-order domain preserving DG method for the two-layer shallow water equations

The bilayer shallow water wave equations in one-dimensional space are considered in this paper. The equations admit two groups of characteristic velocities, which are the first-order approximation of the eigenvalues. Due to the numerical instability, the characteristic velocities may become complex, and thus the system is not hyperbolic and yields to the so-called Kelvin-Helmholtz instability at the interface of the two layers. To overcome this issue, an invariant domain preserving DG method is presented for the bilayer shallow water wave equations. The proposed method is high-order accurate, conservative and can keep the characteristic velocities being real provided that the initial characteristic velocities are real. Therefore, the Kelvin-Helmholtz instability at the interface can be avoided. Representative numerical examples are chose to demonstrate the performance of the proposed method.