Stability and Order of Accuracy Analysis of High-Order Schemes Formulated Using the Flux Reconstruction Approach

TL;DR

This paper analyzes the stability and accuracy of high-order flux reconstruction schemes for the 1D advection equation, revealing that higher order schemes have reduced stability domains and artificial dissipation.

Contribution

It provides a comprehensive stability analysis of high-order FR schemes, including well-known methods like DG, across various spatial discretization orders.

Findings

Higher order schemes have smaller stability domains.

Increased order reduces artificial dissipation.

Stability behavior varies with correction functions.

Abstract

A stability analysis is performed on high-order schemes formulated using the Flux Reconstruction (FR) approach. The one-dimensional advection model equation is used for the assessment of the stability region of these schemes when coupled with Runge-Kutta-type time-march procedures. Schemes are created using different numbers of internal points for each cell of the discrete domain, in such a way that a broad spectrum of spatial discretization orders can be analyzed. Multiple correction functions are employed so that well-known schemes, for instance the Discontinuous Galerkin (DG) and the Staggered-Grid schemes, may also be included in the study. The stability analysis is performed by identifying the behavior, in the complex plane, of the eigenvalues associated with each one of the considered cases. It is observed that, as the order of the scheme increases, a significant decrease in its domain of stability occurs, followed by a significant reduction in the amount of artificial dissipation that is introduced to the solution.