Abstract
This work compares the wave propagation properties of discontinuous Galerkin (DG) schemes for advection–diffusion problems with respect to the behavior of classical discretizations of the diffusion terms, that is, two versions of the local discontinuous Galerkin (LDG) scheme as well as the BR1 and the BR2 scheme. The analysis highlights a significant difference between the two possible ways to choose the alternating LDG fluxes showing that the variant that is inconsistent with the upwind advective flux is more accurate in case of advection–diffusion discretizations. Furthermore, whereas for the BR1 scheme used within a third order DG scheme on Gauss‐Legendre nodes, a higher accuracy for well‐resolved problems has previously been observed in the literature, this work shows that higher accuracy of the BR1 discretization only holds for odd orders of the DG scheme. In addition, this higher accuracy is generally lost on Gauss–Legendre–Lobatto nodes.
Highlights
The wave propagation properties of discontinuous Galerkin (DG) schemes[1] for advection–diffusion problems are compared with respect to the behavior of several classical discretizations of the diffusion terms, that is, two versions of the local discontinuous Galerkin (LDG) scheme[2] as well as the first and second method of Bassi and Rebay,[3,4] termed BR1 and BR2, respectively
We study the wave propagation properties of the DG-discretized linear advection–diffusion equation
In order to give an example of the influence of the grid Peclet number on the results of the eigensolution analysis, in particular for higher order DG schemes, Figure 11 shows the eigenmodes for the two alternate variants of the LDG diffusion fluxes within the DG(N = 5) scheme on Gauss–Legendre nodes for Pe∗ = 20 as well as Pe∗ = 100
Summary
The wave propagation properties of discontinuous Galerkin (DG) schemes[1] for advection–diffusion problems are compared with respect to the behavior of several classical discretizations of the diffusion terms, that is, two versions of the local discontinuous Galerkin (LDG) scheme[2] as well as the first and second method of Bassi and Rebay,[3,4] termed BR1 and BR2, respectively. For advection–diffusion problems, an eigenanalysis by Manzanero et al[32] for DG schemes considers the influence of a parameter-dependent Riemann solver for advective terms and the BR1 scheme for viscous terms Their study investigates both the individual contribution of the dissipative mechanisms on the whole range of wave numbers and their combined effect. For a DG scheme of polynomial degree N = 2 on Gauss–Legendre nodes, differences of its wave propagation properties are studied in case of either upwind or central flux for advection as well as either a particular alternate LDG scheme or the BR1 approach for diffusion.
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