Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-21T17:48:48.907Z Has data issue: false hasContentIssue false
  • English
  • Français

AN EQUAL-AREA METHOD FOR SCALAR CONSERVATION LAWS

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  28 May 2012

MARJETA KRAMAR FIJAVŽ ,
MITJA LAKNER  and
MARJETA ŠKAPIN RUGELJ
MARJETA KRAMAR FIJAVŽ
Affiliation:
Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000 Ljubljana, Slovenia (email: [email protected], [email protected], [email protected])
MITJA LAKNER
Affiliation:
Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000 Ljubljana, Slovenia (email: [email protected], [email protected], [email protected])
MARJETA ŠKAPIN RUGELJ*
Affiliation:
Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000 Ljubljana, Slovenia (email: [email protected], [email protected], [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the one-dimensional conservation law. We use a characteristic surface to define a class of functions, within which the integral version of the conservation law is solved in a simple and direct way. A simple algorithm for computing the unique solution is developed. The method uses the equal-area principle and yields the solution for any given time directly.

Keywords

conservation lawequal areacharacteristicsmesh free

MSC classification

Secondary: 35L65: Conservation laws
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 2 , October 2011 , pp. 156 - 170
Copyright
Copyright © Australian Mathematical Society 2012

References

[1]Bressan, A., Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem (Oxford University Press, New York, 2005).Google Scholar
[2]Cristiani, E., de Fabritiis, C. and Piccoli, B., “A fluid dynamic approach for traffic forecast from mobile sensor data”, Comm. Appl. Ind. Math. 1 (2010) 54–71; doi:10.1685/2010CAIM487.Google Scholar
[3]Farjoun, Y. and Seibold, B., “A rarefaction-tracking method for hyperbolic conservation laws”, J. Engrg. Math. 66 (2010) 237–251; doi:10.1007/s10665-009-9338-3.Google Scholar
[4]Garavello, M. and Piccoli, B., Traffic flow on networks (American Institute of Mathematical Sciences, Springfield, MO, 2006).Google Scholar
[5]Godunov, S. K., “A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations”, Mat. Sb. 47 (1959) 271–306.Google Scholar
[6]Golubitsky, M. and Schaeffer, D. G., “Stability of shock waves for a single conservation law”, Adv. Math. 16 (1975) 65–71; doi:10.1016/0001-8708(75)90100-0.CrossRefGoogle Scholar
[7]Holden, H. and Risebro, N. H., Front tracking for hyperbolic conservation laws (Springer, New York, 2009).Google Scholar
[8]Landau, L. D. and Lifshitz, E. M., Fluid mechanics (Pergamon Press, Oxford, 1987).Google Scholar
[9]Lax, P., Hyperbolic systems of conservation laws and the mathematical theory of shock waves (Society of Industrial and Applied Mathematics, Philadelphia, 1973).CrossRefGoogle Scholar
[10]LeVeque, R. J., Numerical methods for conservation laws (Birkhäuser, Basel, 1992).CrossRefGoogle Scholar
[11]LeVeque, R. J., Finite-volume methods for hyperbolic problems (Cambridge University Press, Cambridge, 2004).Google Scholar
[12] R. J. LeVeque et al., Clawpack, http://www.clawpack.org.Google Scholar
[13]Lighthill, M. J. and Whitham, G. B., “On kinematic waves. II. A theory of traffic flow on long crowded roads”, Proc. R. Soc. Lond. A 229 (1955) 317–345; doi:10.1098/rspa.1955.0089.Google Scholar
[14]Logan, J. D., An introduction to nonlinear partial differential equations (John Wiley & Sons, Hoboken, NJ, 2008).Google Scholar
[15]Richards, P. I., “Shock waves on the highway”, Oper. Res. 4 (1956) 42–51; doi:10.1287/opre.4.1.42.CrossRefGoogle Scholar
[16]Rudin, W., Principles of mathematical analysis (McGraw-Hill, Auckland, 1976).Google Scholar
[17]Schaeffer, D. G., “A regularity theorem for conservation laws”, Adv. Math. 11 (1973) 368–386; doi:10.1016/0001-8708(73)90018-2.CrossRefGoogle Scholar
[18] B. Seibold, Particleclaw, http://www.math.temple.edu/∼seibold/research/particleclaw/ .Google Scholar
[19]Whitham, G. B., Linear and nonlinear waves (John Wiley & Sons, New York, 1974).Google Scholar