Convecting reference frames and invariant numerical models

TL;DR

This paper investigates how using convecting reference frames can improve the numerical accuracy of finite difference models for flow simulations, emphasizing the importance of Galilean invariance in discretization schemes.

Contribution

It introduces invariant discretization schemes for Burgers equation and compares their performance with non-invariant methods and existing remedies.

Findings

Invariant schemes improve numerical results for flow simulations.

Discretizations respecting Galilean invariance outperform traditional methods.

Remedies based on moving reference frames are validated and extended.

Abstract

In the recent paper by Bernardini et al. [J. Comput. Phys. 232 (2013), 1-6] the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied. In a simplified investigation carried out using the viscous Burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of Galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow. Here we further discuss this problem and relate it to known results on invariant discretization schemes. Non-invariant and invariant finite difference discretizations of Burgers equation are proposed and compared with the discretization using the remedy proposed by Bernardini et al..