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Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Galilean invariance and the conservative difference schemes for scalar laws | Ran Advances in Difference Equations 2011 2011 53 http www.advancesindifferenceequations.eom content 2011 1 53 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Galilean invariance and the conservative difference schemes for scalar laws Zheng Ran Correspondence zran@staff.shu. edu.cn Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200o72 P. R. China 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. Keywords difference scheme symmetry shock capturing method Springer 1. Introduction 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 1970 Yanenko and Shokin 2 developed a method of differential approximations for the study of the group properties of difference schemes for hyperbolic systems of equations. They used the first differential approximation to perform a group analysis. A more recent series of articles was devoted to the Lie point symmetries of differential difference equations on 3 . 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 . It is well known that as for Navier-Stokes equations the intrinsic symmetries except for the scaling symmetries are just macroscopic consequences of the basic symmetries of Newton s equations governing microscopic molecular motion in classical approximation . Any physical difference scheme should inherit the elementary symmetries at
