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.
Accepté le :
DOI : 10.1051/m2an/2017027
Keywords: Undercompressive shock waves in 2D, hyperbolic-elliptic systems, interface tracking, Finite Volume method
Chalons, Christophe 1 ; Rohde, Christian 2 ; Wiebe, Maria 2
@article{M2AN_2017__51_5_1987_0,
author = {Chalons, Christophe and Rohde, Christian and Wiebe, Maria},
title = {A finite volume method for undercompressive shock waves in two space dimensions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1987--2015},
year = {2017},
publisher = {EDP Sciences},
volume = {51},
number = {5},
doi = {10.1051/m2an/2017027},
mrnumber = {3731557},
zbl = {1397.76086},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2017027/}
}
TY - JOUR AU - Chalons, Christophe AU - Rohde, Christian AU - Wiebe, Maria TI - A finite volume method for undercompressive shock waves in two space dimensions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1987 EP - 2015 VL - 51 IS - 5 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2017027/ DO - 10.1051/m2an/2017027 LA - en ID - M2AN_2017__51_5_1987_0 ER -
%0 Journal Article %A Chalons, Christophe %A Rohde, Christian %A Wiebe, Maria %T A finite volume method for undercompressive shock waves in two space dimensions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1987-2015 %V 51 %N 5 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2017027/ %R 10.1051/m2an/2017027 %G en %F M2AN_2017__51_5_1987_0
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