Space discontinuous Galerkin method for shallow water flows—kinetic and HLLC flux, and potential vorticity generation

Abstract

In this paper, a second order space discontinuous Galerkin (DG) method is presented for the numerical solution of inviscid shallow water flows over varying bottom topography. Novel in the implementation is the use of HLLC and kinetic numerical fluxes 1 in combination with a dissipation operator, applied only locally around discontinuities to limit spurious numerical oscillations. Numerical solutions over (non-)uniform meshes are verified against exact solutions; the numerical error in the L 2 -norm and the convergence of the solution are computed. Bore-vortex interactions are studied analytically and numerically to validate the model; these include bores as "breaking waves" in a channel and a bore traveling over a conical and Gaussian hump. In these complex numerical test cases, we correctly predict the generation of potential vorticity by non-uniform bores. Finally, we successfully validate the numerical model against measurements of steady oblique hydraulic jumps in a channel with a contraction. In the latter case, the kinetic flux is shown to be more robust.

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

“…We also observe how the numerical scheme generates potential vorticity through the passage of a non-uniform bore [31,32]. For smooth is a conserved quantity [32,33]. Hydraulic bores are discontinuities in the flow where energy is dissipated but mass and momentum are conserved.…”

Hydraulic flow through a channel contraction: Multiple steady states

“…The algorithms and codes used have been verified against rotating and nonrotating exact solutions and validated against experiments and bore-vortex interactions in Refs. [16][17][18][19]. We predominantly used grids of 175ϫ 40 elements and ran a few cases with double resolution as a verification.…”

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

“…Significant progress in the application of DG methods to the SWE has been achieved in the last few years [15,16,17,18,19,20,21,22,23,24,25]. However, two issues relevant in many applications, namely preserving steady-states at rest with variable bathymetry and properly handling flooding and drying, have not been addressed in previous work, with the exception of [17] where a moving mesh was used to deal with dry areas in a one-dimensional setting; the extension to two space dimensions does not seem to be straightforward.…”