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Nonlinear Hyperbolic Systems of Conservation Laws and Related Applications

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Evolutionary Equations with Applications in Natural Sciences

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2126))

Abstract

In this course evolution equations defining non-linear hyperbolic conservation laws, some general theory of non-linear systems of conservation laws and solution methods will be presented. The notions of a weak solution and entropy will be introduced. This will lead into an investigation of solutions of the so called Riemann problem. For scalar conservation laws, analytical solutions will be derived using characteristics methods. In general numerical methods are used to solve or simulate such problems. Therefore, ideas guiding the design of numerical schemes for such equations will be discussed. Some numerical schemes for the numerical integration of such initial boundary value problems related to systems of conservation laws will be analyzed. A collection of case studies from application areas like gas dynamics, and networked flow will be used to demonstrate how non-linear hyperbolic conservation laws are used to model, understand and predict the dynamics governing real-world problems.

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Acknowledgements

The author would like to thank the anonymous reviewer for very constructive comments which have tremendously improved the quality of this chapter. The author would also like to thank the following organisations which funded some of the research presented in this chapter as well as the organisation of the workshop where the topic was presented: International Centre for Pure and Applied Mathematics (CIMPA), International Mathematics Union (IMU), London Mathematical Society—African Mathematics Millennium Science Initiative (LMS-AMMSI), International Centre for Theoretical Physics (ICTP), African Institute of Mathematical Sciences (AIMS), the National Research Foundation (NRF) of South Africa (NRF), University of KwaZulu-Natal, Witwatersrand and Stellenbosch.

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  1. Department of Mathematics and Applied Mathematics, University of Pretoria, X20, Hatfield, 0028, South Africa

    Mapundi Kondwani Banda

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  1. Mapundi Kondwani Banda
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Correspondence to Mapundi Kondwani Banda .

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  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    Jacek Banasiak

  2. Département de Mathématiques, UFR des Sciences et Techniques, Besançon, France

    Mustapha Mokhtar-Kharroubi

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Banda, M.K. (2015). Nonlinear Hyperbolic Systems of Conservation Laws and Related Applications. In: Banasiak, J., Mokhtar-Kharroubi, M. (eds) Evolutionary Equations with Applications in Natural Sciences. Lecture Notes in Mathematics, vol 2126. Springer, Cham. https://doi.org/10.1007/978-3-319-11322-7_9

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