An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

Abstract

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017 ] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N = 7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.

Energy-stable staggered schemes for the Shallow Water equations

“…These tests are one of the few for which exact solutions are available and stand for a relevant validation tool in a fully two-dimensional framework, involving dry cells and varying topography. Two classes of exact solutions are available, namely the planar and curved cases, used for example in [9,21,32,42,54] and [9,18,46] respectively. Based on the COMODO benchmark [1], we consider here the planar case, on a computational domain 200km × 200km, in which the bed profile is defined as follows:…”

Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids

“…A symmetry-preserving approximation for the advection term, which also preserves momentum, is constructed using a 'splitting' strategy very similar to the one presented in [48], [47] and [35], and classical results of Tadmor [30]. For the construction of the advection operator ADVEC, we start with a discrete advection operator ADVEC s that works on a scalar field f v .…”

“…Because of the differences between ADVEC s and ADVEC in the input and output spaces, certain interpolations and rotations are necessary which are not necessary in [48], [47] and [35].…”

“…In these papers, mass, momentum and total energy are conserved on curvilinear meshes, but staggering is not applied. The method is extended to shock capturing and positivity preservation in [48].All the existing models in the literature have their own advantages and disadvantages. In general, they are not at the same time applicable for arbitrary discretization orders, sophisticated operators such as the advection operator, or the chain rule, and curvilinear staggered grids.…”

“…In these papers, mass, momentum and total energy are conserved on curvilinear meshes, but staggering is not applied. The method is extended to shock capturing and positivity preservation in [48].…”