A residual based artificialviscosity method for the stabilizationof the shallow water equations

Abstract: The shallow water equations (SWE) are a system of nonlinear hyperbolic partial differential equations, that have been widely used to modelflows in rivers and coastal areas. For many practical purposes, such as flood mitigation and coastal protection, it is extremely important to develop an accurate, efficient, and robust numerical solver. However, it is known that the numerical solution to nonlinear conservation laws including the shallow water equations is highly non-trivial, as it is susceptive to spurious oscillations that form in the proximity of strong gradients. Other difficulties arise from the appearance of dry areas, as well as the existence of steady-state solutions in which the fluxgradients should be balanced by the source term. To overcome these issues, we adopt a diffusion model, built on a high-order, nonlinear, residual-based stabilization technique, using continuous finite elements. It is shown to be well-balanced with respect to rest states, using the so-called hydrostatic reconstruction technique. To make the numerical scheme high-order in time, we use a Runge-Kutta time-stepping algorithm. Finally, the proposed computational method is illustrated on various benchmark tests in one and two dimensions.