A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system

Manuel Castro; Jorge Macías; Carlos Parés

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 1, page 107-127
  • ISSN: 0764-583X

Abstract

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The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to define a suitable numerical scheme with global upwinding, we first consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.

How to cite

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Castro, Manuel, Macías, Jorge, and Parés, Carlos. "A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system." ESAIM: Mathematical Modelling and Numerical Analysis 35.1 (2010): 107-127. <http://eudml.org/doc/197574>.

@article{Castro2010,
abstract = { The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to define a suitable numerical scheme with global upwinding, we first consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system. },
author = {Castro, Manuel, Macías, Jorge, Parés, Carlos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Q-schemes; coupled conservation laws; source terms; 1D shallow water equations; two-layer flows; hyperbolic systems.; first-order upwind scheme; one-dimensional flow of two superposed immiscible layers},
language = {eng},
month = {3},
number = {1},
pages = {107-127},
publisher = {EDP Sciences},
title = {A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system},
url = {http://eudml.org/doc/197574},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Castro, Manuel
AU - Macías, Jorge
AU - Parés, Carlos
TI - A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 1
SP - 107
EP - 127
AB - The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to define a suitable numerical scheme with global upwinding, we first consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.
LA - eng
KW - Q-schemes; coupled conservation laws; source terms; 1D shallow water equations; two-layer flows; hyperbolic systems.; first-order upwind scheme; one-dimensional flow of two superposed immiscible layers
UR - http://eudml.org/doc/197574
ER -

References

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Citations in EuDML Documents

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  1. C. Bourdarias, M. Ersoy, Stéphane Gerbi, Air entrainment in transient flows in closed water pipes : A two-layer approach
  2. Carlos Parés, Manuel Castro, On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems
  3. Carlos Parés, Manuel Castro, On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems
  4. Emmanuel Audusse, Marie-Odile Bristeau, Finite-volume solvers for a multilayer Saint-Venant system
  5. María Luz Muñoz-Ruiz, Carlos Parés, Godunov method for nonconservative hyperbolic systems
  6. Marica Pelanti, François Bouchut, Anne Mangeney, A Roe-type scheme for two-phase shallow granular flows over variable topography
  7. François Bouchut, Tomás Morales de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment
  8. Jorge Balbás, Smadar Karni, A central scheme for shallow water flows along channels with irregular geometry
  9. Jorge Balbás, Gerardo Hernandez-Duenas, A positivity preserving central scheme for shallow water flows in channels with wet-dry states
  10. Jorge Balbás, Smadar Karni, A central scheme for shallow water flows along channels with irregular geometry

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