Accuracy, Stability, and Performance Comparison between the Spectral Difference and Flux Reconstruction Schemes

TL;DR

This paper compares spectral difference and flux reconstruction schemes in high-order discontinuous spectral element flow solvers, highlighting stability and accuracy differences through theoretical analysis and numerical experiments.

Contribution

It provides a novel nonlinear stability analysis and a fair assessment of both schemes, emphasizing the inherent stability advantages of the true spectral difference scheme.

Findings

True spectral difference scheme exhibits enhanced stability without filtering.

Staggered flux points suppress aliasing errors effectively.

The schemes' performance varies in under-resolved turbulent flow simulations.

Abstract

We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation upon which to provide a fair and accurate assessment of these two schemes in terms of accuracy, stability, and performance with special attention to the true spectral difference scheme and the modified spectral difference scheme recovered via the flux reconstruction formulation. Building on previous analysis of the spectral difference and flux reconstruction schemes, we provide a novel nonlinear stability analysis of the spectral difference scheme. Through various numerical experiments, we demonstrate the additional stability afforded by the true, baseline spectral difference scheme without explicit filtering or de-aliasing due to its inherent feature of staggered flux points. This arrangement leads to favorable suppression of aliasing errors and improves stability needed for under-resolved simulations of turbulent flows.