Volume 51, Number 5, September-October 2017
|Page(s)||1987 - 2015|
|Published online||14 November 2017|
A finite volume method for undercompressive shock waves in two space dimensions∗
1 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France.
2 Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
Received: 20 September 2016
Revised: 23 March 2017
Accepted: 27 April 2017
Undercompressive shock waves arise in many physical processes which involve multiple phases. We propose a Finite Volume method in two space dimensions to approximate weak solutions of systems of hyperbolic or hyperbolic-elliptic conservation laws that contain undercompressive shock waves. The method can be seen as a generalization of the spatially one-dimensional and scalar approach in [C. Chalons, P. Engel and C. Rohde, SIAM J. Numer. Anal. 52 (2014) 554–579]. It relies on a moving mesh ansatz such that the undercompressive wave is represented as a sharp interface without any artificial smearing. It is proven that the method is locally conservative and recovers planar traveling wave solutions exactly. To demonstrate the efficiency and reliability of the new scheme we test it on scalar model problems and as an application on compressible liquid-vapour flow in two space dimensions.
Mathematics Subject Classification: 35L65 / 65M12 / 76M25
Key words: Undercompressive shock waves in 2D / hyperbolic-elliptic systems / interface tracking / Finite Volume method
The first author was partially supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. C.R. and M.W. would like to thank the German Research Foundation for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.
© EDP Sciences, SMAI 2017
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