Abstract
In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.
Highlights
Reviewed by: Hendrik Ranocha, King Abdullah University of Science and Technology, Saudi Arabia Francesco Fambri, Max Planck Institute for Plasma Physics (IPP), Germany
In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics
There is a long history using discontinuous Galerkin (DG) methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics
Summary
The first discontinuous Galerkin (DG) type discretisation is either attributed to Reed and Hill in 1973 [1] for an application to steady state scalar hyperbolic linear advection to model neutron transport, or to Nitsche in 1971 [2] who introduced a discontinuous finite element method (FEM) to solve elliptic problems with non-conforming approximation spaces It was a series of papers by Cockburn and Shu et al starting 20 years later [3,4,5,6] that introduced the modern form of the so-called Runge-Kutta DG scheme. It is important to note that these choices involving the element type, basis functions, and approximation of inner products all have a major impact on the performance of the resulting DG scheme in terms of computational complexity and robustness due, e.g., to the presence of spurious oscillations near discontinuities that result in unphysical solution states (like negative density or pressure) or aliasing instabilities. The remainder of this review article gives the answers to why we need novel developments, Section 2, when the novel developments started, Section 3, what the key ideas of these novel strategies are, Section 4, and where there are still open questions toward future research directions, Section 5
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