Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Abstract

In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

“…The extension of the DGSEM methods to simplicial grids (see [11] for an overview) could be implemented with a few minor modifications. Based on earlier work [22], the necessary basis function implementations for arbitrary quadrature points are available but need to be integrated into the Dune-Fem and Dune-Fem-DG framework.…”

“…with J E being a smoothness indicator detailed in (11) and TOL a threshold that can be influenced by the user.…”

“…125 d5 k denotes a scaling factor, h E is the elements diameter, and |e| the area of the intersection between the two elements E and K . Please note the slight derivation from the notation in [17] by denoting the smoothness indicator in (11) with J E instead of S E .…”

On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

“…It is often useful to mimic the same procedure at the discrete level by using summation by parts (SBP) operators, which are constructed specifically to satisfy a discrete analogue of integration by parts. A review of the relevant theory can be found in [16,26,76]. Many classes of numerical methods can be formulated within the SBP framework, including finite difference [75], finite volume [52,53], continuous Galerkin [1,34,35], discontinuous Galerkin [30], and flux reconstruction methods [69].…”

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

“…To mimic this procedure at the semidiscrete level (discrete in space, continuous in time), summation by parts (SBP) operators are used, which provide a discrete analogue of integration by parts. A review of the relevant theory can be found in [28,41,95]. Nowadays, many different schemes have been formulated in the SBP framework, e.g.…”