On an essentially conservative scheme for hyperbolic conservation laws
type: Item
instance of: scholarly article
title: On an essentially conservative scheme for hyperbolic conservation laws
zbMATH Open document ID: 0807.65098
author: Baoxia Jin
published in: Journal of Computational Physics
publication date: 1995-02-28T00:00:00Z
review text: This paper is concerned with the numerical solution of scalar equations and systems of conservation laws. In the first part of the paper the author proposes an essentially conservative second-order scheme, proving that it satisfies the so called total variation bounded (TVB) property. Such a property is a weaker requirement than the total variation diminishing property considered e.g. by \textit{J. B. Goodman} and \textit{R. J. LeVeque} [Math. Comput. 45, 15-21 (1985; Zbl 0592.65058)] which restricts severely the order of convergence of the schemes. For the scalar conservation law, a detailed theoretical study is carried out in Section 2. Next, the author extends the result of the scalar case to systems of conservation laws by using some kind of averaging, such as \textit{P. L. Roe's} averaging [J. Comput. Phys. 43, 357-372 (1981; Zbl 0474.65066)]. Again a second-order essentially conservative scheme is proposed. In the last section the numerical results obtained by solving numerically several problems with the proposed schemes are presented. As scalar problems a linear equation with periodic initial conditions and a shock wave problem for the inviscid Burgers' equation are considered. As systems of equations, Euler's equations of gas dynamics with discontinuous initial values are considered (here an artificial compression technique is introduced in order to improve the resolution of shock waves). It is shown that the numerical solutions provided by the methods follow closely the behaviour of the exact solutions.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 622262
zbMATH Keywords: systems of conservation laws
zbMATH Keywords: essentially conservative second-order scheme
zbMATH Keywords: total variation diminishing property
zbMATH Keywords: convergence
zbMATH Keywords: periodic initial conditions
zbMATH Keywords: shock wave
zbMATH Keywords: Burgers' equation
zbMATH Keywords: Euler's equations
zbMATH Keywords: gas dynamics
zbMATH Keywords: artificial compression technique
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: jcph.1994.1103
OpenAlex ID: W2068859459
DOI: 10.1006/JCPH.1994.1103
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description: scientific article; zbMATH DE number 622262