An L2-Error Estimate of Energy Stable Flux Reconstruction Method

TL;DR

This paper develops an L2-error estimate for the energy stable flux reconstruction method, revealing how parameter choices affect accuracy and stability in high-order numerical schemes for PDEs.

Contribution

It provides the first L2-error analysis for ESFR schemes, identifying factors leading to loss of optimal order in accuracy.

Findings

L2-error estimate derived for ESFR schemes

Exact expression for order loss identified

Parameter influence on stability and accuracy analyzed

Abstract

Energy stable flux reconstruction (ESFR) is a high-order numerical method used for solving partial differential equations in computational fluid dynamics. This method is designed to preserve the energy stability of the underlying partial differential equation system with respect to a broken Sobolev norm. A class of one-parameter ESFR schemes has been identified to be stable for the one-dimensional linear advection equation. This class includes some well-known high-order methods such as the discontinuous Galerkin method and spectral difference method. The main advantage of the energy stable flux reconstruction is to allow for an increase in the maximum admissible time step while retaining the stability and accuracy properties of the underlying scheme. However numerical experiments have shown that beyond a certain value of the parameter, the optimal order of accuracy is lost. This article develops an L2-error estimate for the energy stable flux reconstruction scheme applied to the one-dimensional advection equation and demonstrates the exact expression that contributes towards the loss of the optimal order.