Discrete and continuous scalar conservation laws
Name
Abeyaratne_MMS_2011_final.pdf
Size
1.79 MB
Format
Adobe PDF
Checksum (MD5)
cfded0ccadd498721d427967ba2ed213
Author(s)
Abeyaratne, Rohan
Knowles, James K.
Date Issued
May 2011
Journal
Mathematics and Mechanics of Solids
Publisher
Sage Publications
Citation
Abeyaratne, R., and J. K. Knowles. “Discrete and Continuous Scalar Conservation Laws.” Mathematics and Mechanics of Solids 16, no. 3 (May 1, 2011): 283–303.
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1177/1081286510382821
