Space-time Discontinuous Galerkin Method for Rotating Shallow Water Flows

In the present work, we analyze the rotating shallow water equations including bottom topography using a space-time discontinuous Galerkin finite element method. The method results in non-linear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a stabilization operator only around discontinuities using Krivodonova’s discontinuity detector. The numerical scheme is verified and validated by comparing numerical and exact solutions, and analyzing bore-vortex interactions. We conclude that the method is second order accurate in both space and time for linear polynomials, and correctly captures bore-vortex interactions. Introduction The rotating shallow water equations including topographic effects are a leading order model to study coastal hydrodynamics on several scales including intermediate scale rotational waves (100 km range) and breaking waves on beaches (1 km range). The shallow water model, in the absence of discontinuities, conserves potential vorticity, enstrophy and energy; and captures many interesting natural wave phenomena like flooding and drying at beaches, hurricanes approaching coastal zones and tsunamis. In this paper, we analyze the rotating shallow water model using a space-time discontinuous Galerkin finite element method. This numerical method is originally developed in [1] to model inviscid compressible flows in a time dependent flow domain. In this method, we discretize and solve the shallow water model per space-time finite element locally by establishing the element communication through a numerical HLLC flux in the spatial direction and a numerical upwind flux in the time direction. The numerical discretization results in a set of coupled nonlinear equations which can be solved efficiently and locally, by adding a pseudo-time derivative and integrating them using a Runge-Kutta scheme until the solution reaches steady state in pseudo-time. We will use a multigrid technique [1] for pseudo-time integration scheme, to improve the convergence acceleration. The nonlinear shallow water equations can develop discontinuities in finite time. To limit spurious oscillations around discontinuities, we employ a stabilization operator (see [1]) only around discontinuities with the help of a discontinuity detector as in [2]. New in the present paper are the application of space-time method to rotating shallow water flows and the combination of a stabilization operator with the discontinuity detector. Furthermore, we present the numerical results for linear polynomials, show that the method is second order accurate in space-time and can accurately capture complex bore-vortex interactions. Rotating Shallow Water Equations Rotating shallow water equations including topographic terms can be concisely given in the index notation as ∇ · Fi(U) = Si in Ω, (1) where ∇ = (∂t, ∂x, ∂y) is the differential operator, U = (h, hu, hv) the state vector, h(x) the water depth, (u(x), v(x)) the velocity field,