A comparative Fourier analysis of discontinuous Galerkin schemes for advection–diffusion with respect to BR1, BR2, and local discontinuous Galerkin diffusion discretization

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.

Two-derivative deferred correction time discretization for the discontinuous Galerkin method

“…Here, we choose the BR2 scheme because of its compact stencil as it has only dependencies on direct neighbors, see e.g. [39].…”

Fourier Analysis of DG Schemes for Advection‐Diffusion

“…where ûadv = 1 2 (u − + u + ) + γ 2 (u − − u + ) represents the choice of numerical advection flux of either uwind (γ = 1) or central type (γ = 0) and q and ûdiv denote the numerical diffusion fluxes LDG a , BR1 and BR2(η e = 3) described in detail in [6].…”

“…In this work, the various choices of numerical diffusion fluxes and DG nodal sets are combined with either central or upwind fluxes for advection whereas [6] only considers upwind advection discretization.…”

“…Furthermore, DG schemes on Gauss-Lobatto nodes are considered as well as a larger range of interface fluxes for diffusion terms. In particular, it is shown in [6] that the two possible choices of alternating LDG fluxes lead to significant differences in the error vs. wave number characteristics and that the higher accuracy of the BR1 flux for well-resolved solutions observed in [5] is restricted to the DG scheme on Gauss nodes and to an even polynomial degree N .…”

“…It is shown that the corresponding schemes with central flux discretization produce smaller errors for well-resolved solutions whereas one-sided flux discretizations produce smaller errors in the under-resolved case. In [6], the study by Watkins et al [5] has been extended to higher order DG schemes with N > 2 to detect odd-even phenomena. Furthermore, DG schemes on Gauss-Lobatto nodes are considered as well as a larger range of interface fluxes for diffusion terms.…”