Error boundedness of Correction Procedure via Reconstruction / Flux Reconstruction

TL;DR

This paper investigates the long-term error behavior of CPR/FR methods for linear hyperbolic laws, highlighting the significant impact of basis choice and numerical flux on error growth and asymptotic accuracy.

Contribution

It demonstrates that basis selection (Gau{ extquoteright}ss-Legendre vs. Gau{ extquoteright}ss-Lobatto) critically influences error dynamics, with Gau{ extquoteright}ss-Legendre bases providing faster and lower asymptotic errors.

Findings

Gau{ extquoteright}ss-Legendre basis reduces long-term error more effectively.

Basis choice impacts error growth and asymptotic value.

Numerical flux choice has less effect at low resolutions with Gau{ extquoteright}ss-Legendre basis.

Abstract

We study the long-time error behavior of correction procedure via reconstruction / flux reconstruction (CPR/FR) methods for linear hyperbolic conservation laws. We show that not only the choice of the numerical flux (upwind or central) affects the growth rate and asymptotic value of the error, but that the selection of bases (Gau{\ss}-Lobatto or Gau{\ss}-Legendre) is even more important. Using a Gau{\ss}-Legendre basis, the error reaches the asymptotic value faster and to a lower value than when using a Gau{\ss}-Lobatto basis. Also, the differences in the error caused by the numerical flux are not essential for low resolution computations in the Gau{\ss}-Legendre case. This behavior is better seen on a particular FR scheme which has a strong connection with the discontinuous Galerkin framework but holds also for other flux reconstruction schemes with low order resolution computations.