Central-Upwind Schemes for the Saint-Venant System

Alexander Kurganov; Doron Levy

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 3, page 397-425
  • ISSN: 0764-583X

Abstract

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We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.

How to cite

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Kurganov, Alexander, and Levy, Doron. "Central-Upwind Schemes for the Saint-Venant System." ESAIM: Mathematical Modelling and Numerical Analysis 36.3 (2010): 397-425. <http://eudml.org/doc/194110>.

@article{Kurganov2010,
abstract = { We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas. },
author = {Kurganov, Alexander, Levy, Doron},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Saint-Venant system; shallow water equations; high-order central-upwind schemes; balance laws conservation laws; source terms.; high-order central-upwind schemes; balance laws; conservation laws; source terms},
language = {eng},
month = {3},
number = {3},
pages = {397-425},
publisher = {EDP Sciences},
title = {Central-Upwind Schemes for the Saint-Venant System},
url = {http://eudml.org/doc/194110},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Kurganov, Alexander
AU - Levy, Doron
TI - Central-Upwind Schemes for the Saint-Venant System
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 3
SP - 397
EP - 425
AB - We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.
LA - eng
KW - Saint-Venant system; shallow water equations; high-order central-upwind schemes; balance laws conservation laws; source terms.; high-order central-upwind schemes; balance laws; conservation laws; source terms
UR - http://eudml.org/doc/194110
ER -

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Citations in EuDML Documents

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  1. Steve Bryson, Yekaterina Epshteyn, Alexander Kurganov, Guergana Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
  2. Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto, An entropy-correction free solver for non-homogeneous shallow water equations
  3. Smadar Karni, Eduard Kirr, Alexander Kurganov, Guergana Petrova, Compressible two-phase flows by central and upwind schemes
  4. Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto, An entropy-correction free solver for non-homogeneous shallow water equations
  5. Smadar Karni, Eduard Kirr, Alexander Kurganov, Guergana Petrova, Compressible two-phase flows by central and upwind schemes
  6. Steve Bryson, Yekaterina Epshteyn, Alexander Kurganov, Guergana Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
  7. Alina Chertock, Alexander Kurganov, On a hybrid finite-volume-particle method
  8. Alina Chertock, Alexander Kurganov, On a hybrid finite-volume-particle method
  9. Jorge Balbás, Smadar Karni, A central scheme for shallow water flows along channels with irregular geometry
  10. Jorge Balbás, Gerardo Hernandez-Duenas, A positivity preserving central scheme for shallow water flows in channels with wet-dry states

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