On the Superconvergence of ESFR Schemes

TL;DR

This paper provides a theoretical proof for the superconvergence properties of ESFR schemes in linear advection, explaining their high accuracy and validating the results through numerical experiments.

Contribution

It offers the first formal derivation of superconvergence in ESFR schemes, linking it to rational approximants of the exponential function and analyzing accuracy drops at high scalar values.

Findings

Superconvergence relies on rational approximants of the exponential function.

Accuracy drops are due to changes in the structure of rational approximants and eigenvalue multiplicity.

Theoretical results are validated by numerical experiments.

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.