We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ``black-box’’ solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i) to show that the main drawback of the original central-upwind approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account and (ii) to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. We apply the new scheme to the Saint-Venant system with discontinuous bottom topography and two-layer shallow water system. Our numerical results illustrate the good performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution.
Keywords: Nonconservative hyperbolic systems of PDEs, Saint-Venant system, two-layer shallow water equations, central-upwind scheme, path-conservative scheme, well-balanced scheme
Castro Díaz, Manuel Jesús 1 ; Kurganov, Alexander 1 ; Morales de Luna, Tomás 1
@article{M2AN_2019__53_3_959_0,
author = {Castro D{\'\i}az, Manuel Jes\'us and Kurganov, Alexander and Morales de Luna, Tom\'as},
title = {Path-conservative central-upwind schemes for nonconservative hyperbolic systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {959--985},
year = {2019},
publisher = {EDP Sciences},
volume = {53},
number = {3},
doi = {10.1051/m2an/2018077},
zbl = {1418.76034},
mrnumber = {3969161},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2018077/}
}
TY - JOUR AU - Castro Díaz, Manuel Jesús AU - Kurganov, Alexander AU - Morales de Luna, Tomás TI - Path-conservative central-upwind schemes for nonconservative hyperbolic systems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 959 EP - 985 VL - 53 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2018077/ DO - 10.1051/m2an/2018077 LA - en ID - M2AN_2019__53_3_959_0 ER -
%0 Journal Article %A Castro Díaz, Manuel Jesús %A Kurganov, Alexander %A Morales de Luna, Tomás %T Path-conservative central-upwind schemes for nonconservative hyperbolic systems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 959-985 %V 53 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2018077/ %R 10.1051/m2an/2018077 %G en %F M2AN_2019__53_3_959_0
Castro Díaz, Manuel Jesús; Kurganov, Alexander; Morales de Luna, Tomás. Path-conservative central-upwind schemes for nonconservative hyperbolic systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 959-985. doi: 10.1051/m2an/2018077
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