Generalised Sobolev Stable Flux Reconstruction

TL;DR

This paper introduces a new set of symmetric correction functions for high-order flux reconstruction that unify previous methods, improve stability, and enhance performance for linear and non-linear equations.

Contribution

It develops a generalized framework for correction functions in flux reconstruction, expanding the stability and applicability of the method.

Findings

New correction functions extend temporal stability limits.

Legendre polynomials used to realize higher-order correction functions.

Improved performance in aliasing and non-linear equation tests.

Abstract

A new set of symmetric correction functions is presented for high-order flux reconstruction, that expands upon, while incorporating, all previous correction function sets and opens the possibility for improved performance. By considering FR applied to the linear advection equation, and through modification to the Sobolev norm, criteria are presented for a wider set of correction functions. Legendre polynomials are then used to fulfil these criterion and realise functions for third to fifth order FR. The sufficient conditions for the existence of the modified norms are also explored, before Fourier and Von Neumann analysis are applied to analytically find temporal stability limits for various Runge-Kutta temporal integration schemes. For all cases, correction functions are found that extend the temporal stability of FR. Two application-inspired investigations are performed that aim to explore the effect of aliasing and non-linear equations. In both cases unique correction functions could be found that give good performance, compared to previous FR schemes, while also improving upon the temporal stability limit.