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Linearly Stable Generalizations of ESFR Schemes

Mathias Dufresne-Piché, Siva Nadarajah

TL;DR

This work introduces Sobolev Stable DG (SSDG) as a conservative, linearly stable FDG generalization of Energy Stable Flux Reconstruction (ESFR) schemes and situates it relative to Extended ESFR (EESFR) and Generalized Sobolev FR (GSFR). Using Von Neumann analysis, the authors compare SSDG and EESFR in terms of dispersion, dissipation, spectral accuracy, and CFL limits, showing that SSDG can achieve comparable CFL gains to EESFR but over a much larger parameter range, while preserving p-order accuracy under certain limits. SSDG’s two-parameter construction yields a diagonal, positive-definite stabilization that can reduce to ESFR$p-1$ in one limit and to DG$(p-2)$ in another, offering wide practical flexibility. The paper also highlights that SSDG is easier to implement in DG codes, naturally extends to triangular/tetrahedral elements, and remains NSFR-compatible, making it attractive for high-order CFD on complex meshes and potential shock-capturing applications.

Abstract

The energy stable flux reconstruction (ESFR) method encompasses an infinite family of high-order, linearly stable schemes and thus provides a flex- ible and efficient framework for achieving high levels of accuracy on unstruc- tured grids. One remarkable property of ESFR schemes is their ability to be expressed equivalently as linearly filtered discontinuous Galerkin (FDG) schemes. In this study, we introduce Sobolev Stable discontinuous Galerkin (SSDG) schemes, a new conservative and linearly stable generalization of ESFR schemes via the FDG framework. Additionally, we review existing generalizations of the ESFR method and consider their relationship with the FDG framework. The linear properties of SSDG schemes are studied via Von Neumann analysis and compared to those of the existing extended ESFR (EESFR) method. It is found that while SSDG and EESFR schemes exhibit fundamentally different dispersive and dissipative behaviors, they can achieve a similar increase in CFL limit and exhibit a similar spectral order of accuracy. Moreover, it is seen that the range of scheme parameters over which SSDG schemes can be used to increase the explicit time-stepping limit is much larger than for EESFR schemes. Finally, it is observed that the order of accuracy of EESFR schemes under h-refinement is generally p + 1 while that of SSDG schemes is p.

Linearly Stable Generalizations of ESFR Schemes

TL;DR

This work introduces Sobolev Stable DG (SSDG) as a conservative, linearly stable FDG generalization of Energy Stable Flux Reconstruction (ESFR) schemes and situates it relative to Extended ESFR (EESFR) and Generalized Sobolev FR (GSFR). Using Von Neumann analysis, the authors compare SSDG and EESFR in terms of dispersion, dissipation, spectral accuracy, and CFL limits, showing that SSDG can achieve comparable CFL gains to EESFR but over a much larger parameter range, while preserving p-order accuracy under certain limits. SSDG’s two-parameter construction yields a diagonal, positive-definite stabilization that can reduce to ESFR in one limit and to DG in another, offering wide practical flexibility. The paper also highlights that SSDG is easier to implement in DG codes, naturally extends to triangular/tetrahedral elements, and remains NSFR-compatible, making it attractive for high-order CFD on complex meshes and potential shock-capturing applications.

Abstract

The energy stable flux reconstruction (ESFR) method encompasses an infinite family of high-order, linearly stable schemes and thus provides a flex- ible and efficient framework for achieving high levels of accuracy on unstruc- tured grids. One remarkable property of ESFR schemes is their ability to be expressed equivalently as linearly filtered discontinuous Galerkin (FDG) schemes. In this study, we introduce Sobolev Stable discontinuous Galerkin (SSDG) schemes, a new conservative and linearly stable generalization of ESFR schemes via the FDG framework. Additionally, we review existing generalizations of the ESFR method and consider their relationship with the FDG framework. The linear properties of SSDG schemes are studied via Von Neumann analysis and compared to those of the existing extended ESFR (EESFR) method. It is found that while SSDG and EESFR schemes exhibit fundamentally different dispersive and dissipative behaviors, they can achieve a similar increase in CFL limit and exhibit a similar spectral order of accuracy. Moreover, it is seen that the range of scheme parameters over which SSDG schemes can be used to increase the explicit time-stepping limit is much larger than for EESFR schemes. Finally, it is observed that the order of accuracy of EESFR schemes under h-refinement is generally p + 1 while that of SSDG schemes is p.
Paper Structure (23 sections, 5 theorems, 63 equations, 16 figures, 1 table)

This paper contains 23 sections, 5 theorems, 63 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Filtered Nodal Discontinuous Galerkin
  4. Linear Stability and Conservativeness
  5. Flux Reconstruction
  6. Conservativeness
  7. Energy Stable Flux Reconstruction Schemes
  8. Generalizations of ESFR Schemes
  9. Extended Energy Stable Flux Reconstruction Schemes
  10. Sobolev Stable Discontinuous Galerkin Schemes
  11. Generalized Sobolev Stable Flux Reconstruction
  12. Connections between Filtered DG and FR Schemes
  13. Von Neumann Analysis
  14. Spectral Error
  15. Stability
  16. ...and 8 more sections

Key Result

Theorem 1

The spatial discretization given by Eq.(eq:FDG_matrix_linear) is stable provided that $\mathbf{M} + \mathbf{K} \succ 0$ and the numerical flux is given by where $u^+$ and $u^-$ represent the solution values at the right and left of a given elemental interface and $\alpha \in [0,1]$.

Figures (16)

  • Figure 1: Relationship between FR, FDG, EESFR, SSDG and ESFR schemes.
  • Figure 2: Dispersion relations for two-parameter EESFR and SSDG schemes with $p=3$, $c_{p} = c_{DG}$ and $q_0 = k_p^2 c_{DG}$ for an upwind numerical flux.
  • Figure 3: Dissipation relations for two-parameter EESFR and SSDG schemes with $p=3$, $c_{p} = c_{DG}$ and $q_0 = k_p^2 c_{DG}$ for an upwind numerical flux.
  • Figure 4: Dispersion relations for two-parameter EESFR and SSDG schemes with $p=3$, $c_{p} = c_{SD}$ and $q_0 = k_p^2 c_{SD}$ for an upwind numerical flux.
  • Figure 5: Dissipation relations for two-parameter EESFR and SSDG schemes with $p=3$, $c_{p} = c_{SD}$ and $q_0 = k_p^2 c_{SD}$ for an upwind numerical flux.
  • ...and 11 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Theorem 4
  • proof