Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws

Abstract

Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG methods. We present several implementation techniques to improve the efficiency of flux differencing DG methods that use tensor product quadrilateral or hexahedral elements, in 2D or 3D respectively. Focus is mostly given to CPUs and DG methods for the compressible Euler equations, although these techniques are generally also useful for other physical systems including the compressible Navier-Stokes and magnetohydrodynamics equations. We present results using two open source codes, Trixi.jl written in Julia and FLUXO written in Fortran, to demonstrate that our proposed implementation techniques are applicable to different code bases and programming languages.

References in corpus (4)

• Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing
• An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing
• Entropy-stable, high-order summation-by-parts discretizations without interface penalties
• On the implementation of a robust and efficient finite element-based parallel solver for the compressible Navier-Stokes equations

Cited by in corpus (5)

• Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing
• Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra
• On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws
• Finite-difference compatible entropy-conserving schemes for the compressible Euler equations
• Multiderivative time integration methods preserving nonlinear functionals via relaxation