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Discrete and continuous scalar conservation laws

Abeyaratne, Rohan and Knowles, James K. (2011) Discrete and continuous scalar conservation laws. Mathematics and Mechanics of Solids, 16 (3). pp. 283-303. ISSN 1081-2865.

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Motivated by issues arising for discrete second-order conservation laws and their continuum limits (applicable, for example, to one-dimensional nonlinear spring—mass systems), here we study the corresponding issues in the simpler setting of first-order conservation laws (applicable, for example, to the simplest theory of traffic flow). The discrete model studied here comprises a system of first-order nonlinear differential-difference equations; its continuum limit is a one-dimensional scalar conservation law. Our focus is on issues related to discontinuities — shock waves — in the continuous theory and the corresponding regions of rapid change in the discrete model. In the discrete case, we show that a family of new conservation laws can be deduced from the basic one, while in the continuous case we show that this is true only for smooth solutions. We also examine how well the continuous model approximates rapidly changing solutions of the discrete model, and this leads us to derive an improved continuous model which is of second-order. We also consider the form and implications of the second law of thermodynamics at shock waves in the scalar case.

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Additional Information:© 2010 The Author(s). Received 22 April 2010; accepted 27 July 2010. Published online before print November 24, 2010. The authors wish to thank Srikanth Vedantam for his help with some Mathematica calculations and Debbie Blanchard for her help with the figures. The work described in this paper, and a preliminary version of the manuscript, were completed before the passing away of Jim Knowles in November 2009. Some final changes to the manuscript were subsequently made by Rohan Abeyaratne. Funding: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit scctors.
Subject Keywords:discrete conservation laws, continuous conservation laws, field equations, shock waves, dissipation, entropy
Issue or Number:3
Record Number:CaltechAUTHORS:20110621-110113923
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Official Citation: Rohan Abeyaratne and James K. Knowles Discrete and continuous scalar conservation laws Mathematics and Mechanics of Solids May 2011 16: 283-303, first published on November 24, 2010 doi:10.1177/1081286510382821
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:24152
Deposited By: Ruth Sustaita
Deposited On:21 Jun 2011 18:25
Last Modified:03 Oct 2019 02:53

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