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On the Superconvergence of ESFR Schemes

Mathias Dufresne-Piché, Siva Nadarajah

TL;DR

This work provides a formal proof of the superconvergent dispersion-dissipation error for energy stable flux reconstruction (ESFR) schemes in the linear advection setting with an upwind flux. By linking the error to the determinant of a modified von Neumann matrix and to superconvergent rational approximants of $e^{-z}$, it explains how the ESFR parameter $c$ shapes the observed convergence rates and reconciles them with the DG limit as $c\to\infty$. The analysis yields explicit expressions involving $(p+1,p)$ Padé-like approximants and clarifies why large $c$ can induce apparent rate drops due to unphysical eigenvalues, with numerical experiments confirming the theory. The results extend to a broader class of symmetric FR schemes and offer a principled framework for understanding and predicting long-time accuracy of ESFR-based discretizations in CFD contexts.

Abstract

The energy stable flux reconstruction (ESFR) method provides an efficient and flexible framework to devise high-order linearly stable numerical schemes which can achieve high levels of accuracy on unstructured grids. While superconvergent properties of ESFR schemes have been observed in numerical experiments, no formal proof of this behavior has been reported in the literature. In this work, we attempt to address this by providing a simple derivation for the superconvergence of the dispersion-dissipation error of ESFR schemes for the linear advection problem when using an upwind numerical flux. We show that the superconvergence of ESFR schemes essentially relies on the capacity of the latter to generate superconvergent rational approximants of the exponential function, which is reminiscent of well-known theoretical results for superconvergence of discontinuous Galerkin (DG) methods. We also demonstrate that the drops in order of accuracy which are observed in numerical experiments as the ESFR scalar $c$ is increased are caused by both a modification of the structure of these rational approximants and a change in the multiplicity of the physical eigenvalue of the schemes as $c \to \infty$. Finally, our theoretical results are successfully validated against numerical experiments.

On the Superconvergence of ESFR Schemes

TL;DR

This work provides a formal proof of the superconvergent dispersion-dissipation error for energy stable flux reconstruction (ESFR) schemes in the linear advection setting with an upwind flux. By linking the error to the determinant of a modified von Neumann matrix and to superconvergent rational approximants of , it explains how the ESFR parameter shapes the observed convergence rates and reconciles them with the DG limit as . The analysis yields explicit expressions involving Padé-like approximants and clarifies why large can induce apparent rate drops due to unphysical eigenvalues, with numerical experiments confirming the theory. The results extend to a broader class of symmetric FR schemes and offer a principled framework for understanding and predicting long-time accuracy of ESFR-based discretizations in CFD contexts.

Abstract

The energy stable flux reconstruction (ESFR) method provides an efficient and flexible framework to devise high-order linearly stable numerical schemes which can achieve high levels of accuracy on unstructured grids. While superconvergent properties of ESFR schemes have been observed in numerical experiments, no formal proof of this behavior has been reported in the literature. In this work, we attempt to address this by providing a simple derivation for the superconvergence of the dispersion-dissipation error of ESFR schemes for the linear advection problem when using an upwind numerical flux. We show that the superconvergence of ESFR schemes essentially relies on the capacity of the latter to generate superconvergent rational approximants of the exponential function, which is reminiscent of well-known theoretical results for superconvergence of discontinuous Galerkin (DG) methods. We also demonstrate that the drops in order of accuracy which are observed in numerical experiments as the ESFR scalar is increased are caused by both a modification of the structure of these rational approximants and a change in the multiplicity of the physical eigenvalue of the schemes as . Finally, our theoretical results are successfully validated against numerical experiments.

Paper Structure

This paper contains 15 sections, 6 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ESFR schemes for the Linear Advection Problem
  4. Dispersion-dissipation Error of ESFR Schemes
  5. Proof for Superconvergence Property
  6. Equivalent Problem
  7. Superconvergence of the Dispersion-Dissipation Error
  8. Important Remarks
  9. Numerical Experiments
  10. Convergence of the Dispersion-Dissipation Error
  11. Validity of the Superconvergent Error Estimate
  12. Extension to General Symmetric FR Schemes
  13. Conclusion
  14. Algebraic Multiplicity of Null Eigenvalue
  15. Symmetric FR Schemes

Key Result

Theorem 3.1

Let $\tilde{\mathbf{H}} : \mathbb{R} \to \mathbb{C}^{(p+1)\times(p+1)}$ be defined as Then, as $\theta \to 0$, $E_T(\theta)$ and $|\text{det}(\tilde{\mathbf{H}}(\theta))|$ converge to 0 at the same rate provided that the ESFR parameter satisfies $c_{-} < c < \infty$.

Figures (2)

  • Figure 1: Convergence rate of $E_T$ and $|\lambda_1(c,\theta)-\theta|$ for $p=2$ and $c=100$.
  • Figure 2: Experimental validation of the error estimate.

Theorems & Definitions (13)

  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Lemma 3.3
  • Proof 2
  • Lemma 3.4
  • Proof 3
  • Theorem 3.5
  • Proof 4
  • Lemma A.1
  • ...and 3 more