Galilean invariance and the conservative difference schemes for scalar laws

Zheng Ran

doi:10.1186/1687-1847-2011-53

Abstract

Galilean invariance for general conservative finite difference schemes is presented in this article. Two theorems have been obtained for first- and second-order conservative schemes, which demonstrate the necessity conditions for Galilean preservation in the general conservative schemes. Some concrete application has also been presented.

Highlights

For gas dynamics, the non-invariance relative to Galilean transformation of a difference scheme which approximates the equations results in non-physical fluctuations, that has been marked in the 1960s of the past century [1]
In a series of more recent articles, the author of this article has used Lie symmetry analysis method to investigate some noteworthy properties of several difference schemes for nonlinear equations in shock capturing [4,5]
Any physical difference scheme should inherit the elementary symmetries from the Navier-Stokes equations. This means that Galilean invariance has been an important issue in computational fluid dynamics (CFD)

Summary

Introduction

The non-invariance relative to Galilean transformation of a difference scheme which approximates the equations results in non-physical fluctuations, that has been marked in the 1960s of the past century [1]. The symmetry group of a system of differential equations is the largest local group of transformations acting on the independent and dependent variables of the system such that it can transform one system solution to another. The search for the symmetry algebra L of a system of differential equations is best formulated in terms of vector fields acting on the space X × U of independent and dependent variables. We need to know how the derivatives, that is ux, uxx,..., transform This is given by the prolongation of the vector field V. This theorem is used to deduce explicitly different infinitesimal conditions for specific problems It must be remembered, that, in all cases, though only the scalar differential problem is being discussed, Δv is still used to denote different differential equations

Galilean group and its prolongation

General 3-point conservation scheme

Conclusions