Geodesic
A finite volume method for undercompressive shock waves in two space dimensions
Chalons, Christophe1  ; Rohde, Christian2  ; Wiebe, Maria2
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.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1987-2015

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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.

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DOI : 10.1051/m2an/2017027
Classification : 35L65, 65M12, 76M25
Keywords: Undercompressive shock waves in 2D, hyperbolic-elliptic systems, interface tracking, Finite Volume method

Chalons, Christophe 1 ; Rohde, Christian 2 ; Wiebe, Maria 2

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. 
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     title = {A finite volume method for undercompressive shock waves in two space dimensions},
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Chalons, Christophe; Rohde, Christian; Wiebe, Maria. A finite volume method for undercompressive shock waves in two space dimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1987-2015. doi: 10.1051/m2an/2017027

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