Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods

Jan Glaubitz, Hendrik Ranocha, Andrew R. Winters, Michael Schlottke-Lakemper, Philipp Öffner, Gregor Gassner

Abstract

High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously -- sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin--Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes...

Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods

Abstract

High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously -- sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin--Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes...
Paper Structure (32 sections, 5 theorems, 63 equations, 5 figures, 6 tables)

This paper contains 32 sections, 5 theorems, 63 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on SBP and USBP operators
  3. Notation
  4. SBP operators
  5. USBP operators
  6. Characterizing the existence of USBP operators
  7. Constructing USBP operators
  8. Proposed procedure
  9. Constructing a dissipation matrix
  10. Robust implementation using discrete orthogonal polynomials
  11. An example: USBP operators on LGL points
  12. Application of USBP operators to nodal DG methods
  13. The flux vector splitting approach
  14. Application to nodal collocated DG methods
  15. Numerical tests
  16. ...and 17 more sections

Key Result

Lemma 2.3

\newlabellem:connection0 If $D_{\pm} = P^{-1}( Q_{\pm} + B/2 )$ are degree $d$ USBP operators, then is a degree $d$ SBP operator. Moreover, $D_+ - D_- = P^{-1} S$.

Figures (5)

  • Figure 1: Spectra of the DG-USBP and the DGSEM semi-discretizations with $J=16$ elements. We use degree two ($d=2$, \ref{['fig:spectra_N4_USBP_vs_DGSEM_16elements']}) and three ($d=3$, \ref{['fig:spectra_N5_USBP_vs_DGSEM_16elements']}) DG-USBP operators on four ($N=4$) and five ($N=5$) LGL nodes, respectively, with different parameters $\lambda_4$ and $\lambda_5$. For the DGSEM method, we use traditional degree three and four DG-SBP operators on the same LGL nodes as the respective DG-USBP operators.
  • Figure 2: Meshes of 550 non-overlapping quadrilaterals with linear and quartic boundary polynomial order for the FSP test. Observe---see the lower right corner---that the boundary is approximated more accurately for the higher (quartic) boundary polynomial.
  • Figure 3: Density error for the isentropic vortex test case of the DG-USBP and DGSEM method for $J=256$ elements. We use degree one ($d=1$, blue dashed line), two ($d=2$, orange dotted line), and three ($d=3$, green dash-dotted line) tensor-product DG-USBP operators on $N=3, 4, 5$ LGL nodes per coordinate direction and element. For the DGSEM method, we use traditional degree two, three, and four DG-SBP operators on the same LGL nodes as the DG-USBP operators.
  • Figure 4: Density profile for the DG-USBP method and DGSEM on $J=1024$ elements when the Kelvin--Helmholtz instability simulations crashed. Both methods use six LGL nodes per coordinate direction and element. Hence, all simulations use the same number of DOFs. Furthermore, the DG-USBP methods use $\lambda_5 = -10^{-1}$. The white spots mark points where the pressure (DG-USBP) or the density (DGSEM) is non-positive.
  • Figure 5: Discrete kinetic energy and dissipation rate for the inviscid Taylor--Green vortex. We compare the results of a DG-USBP method with the DGSEM. Both methods use $J=4096$ elements with three ($N=3$, top row) and four ($N=4$, bottom row) LGL points per element and coordinate direction. The DGSEM uses the entropy-conservative volume flux of Ranocha and the local Lax-Friedrichs (Rusanov) surface flux. The DG-USBP method uses the Steger--Warming splitting and $\lambda_{N} = -10^{-1}$.

Theorems & Definitions (15)

  • Definition 2.1: SBP operators
  • Definition 2.2: USBP operators
  • Lemma 2.3: USBP operators induce SBP operators
  • Proof 1
  • Theorem 3.1
  • Lemma 3.2
  • Proof 2
  • Remark 4.1
  • Remark 4.2: Finding DOP bases
  • Remark 5.1
  • ...and 5 more