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

Manuel Castro; Jorge Macías; Carlos Parés

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • 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, 26, 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 shadow water system." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.1 (2001): 107-127. <http://eudml.org/doc/194038>.

@article{Castro2001,
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, 26, 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 - Modélisation Mathématique et Analyse Numérique},
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},
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 shadow water system},
url = {http://eudml.org/doc/194038},
volume = {35},
year = {2001},
}

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 shadow water system
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
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, 26, 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/194038
ER -

References

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  1. [1] L. Armi, The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163 (1986) 27–58. 
  2. [2] L. Armi and D. Farmer, Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164 (1986) 27–51. Zbl0587.76168
  3. [3] A. Bermúdez and M.E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms. Computers and Fluids 23 (1994) 1049–1071. Zbl0816.76052
  4. [4] C. Berthon and F. Coquel, Travelling wave solutions of a convective diffusive system with first and second order terms in nonconservation form, in Hyperbolic Problems: Theory, Numerics, Applications, Vol. I of Internat. Ser. Numer. Math. 129, Birkhäuser (1999) 47–54. Zbl0934.35030
  5. [5] M.J. Castro, J. Macías and C. Parés, Simulation of two-layer exchange flows through a contraction with a finite volume shallow water model, in Actas de las II Jornadas de Análisis de Variables y Simulación Numérica del Intercambio de Masas de Agua a través del Estrecho de Gibraltar, Cádiz (2000) 205–221. 
  6. [6] M.J. Castro, J. Macías and C. Parés, Simulation of two-layer exchange flows through the combination of a sill and contraction with a finite volume shallow-water model. Internal Journal 1610, group on “Differential Equations, Numerical Analysis and Applications”, University of Málaga (2000). 
  7. [7] F. Coquel, K. El Amine, E. Godlewski, B. Perthame and P. Rascle, Une méthode numérique decentrée pour la résolution d’ecoulements diphasiques. C. R. Acad. Sci. Paris, Sér. I 324 (1997) 717–723. Zbl0879.76057
  8. [8] S.B. Dalziel, Two-layer Hydraulics Maximal Exchange Flows. Ph.D. thesis, University of Cambridge (1988). 
  9. [9] D. Farmer and L. Armi, Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow. J. Fluid Mech. 164 (1986) 53–76. Zbl0587.76169
  10. [10] P. García-Navarro and F. Alcrudo, Implicit and explicit TVD methods for discontinuous open channel flows, in Proc. of the 2nd Int. Conf. on Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters, R.A. Falconer, K. Shiono, and R.G.S. Matthew, Eds. 2 Ashgate (1992). 
  11. [11] P. García-Navarro, F. Alcrudo and J.M. Savirón, 1D open channel flow simulation using TVD McCormack scheme. J. Hydraul. Eng. 118 (1992) 1359–1373. 
  12. [12] A.E. Gill, Atmosphere-Ocean Dynamics, Int. Geophys. Series 30, Springer-Verlag, San Diego (1982) 662 p. 
  13. [13] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Sc. 118, Springer-Verlag, New York (1996). Zbl0860.65075MR1410987
  14. [14] A. Harten, On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21 (1984) 1–23. Zbl0547.65062
  15. [15] A. Harten, P. Lax and A. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. Zbl0565.65051
  16. [16] K.R. Helfrich, Time-dependent two-layer hydraulic exchange flows. J. Phys. Oceanogr. 25 (1995) 359–373. 
  17. [17] P.K. Kundu, Fluid Mechanics. Academic Press Inc., San Diego (1990) 638 p. Zbl0780.76001
  18. [18] P.G. Lefloch and A.E. Tzavaras, Existence theory for the Riemann problem for non-conservative hyperbolic systems. C. R. Acad. Sci. Paris, Sér. I 323 (1996) 347–352. Zbl0864.35073
  19. [19] P.G. Lefloch and A.E. Tzavaras, Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30 (1999) 1309–1342. Zbl0939.35115
  20. [20] P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357–371. Zbl0474.65066
  21. [21] P.L. Roe, Upwinding differenced schemes for hyperbolic conservation laws with source terms, in Proc. of the Conference on Hyperbolic Problems, C. Carasso, P.-A. Raviart, and D. Serre, Eds., Springer-Verlag, Berlin (1986) 41–51. Zbl0626.65086
  22. [22] J.A. Rubal and M.E. Vázquez, Aplicación del método de volúmenes finitos y esquemas tipo Godunov a un modelo bicapa, in Actas de las II Jornadas de Análisis de Variables y Simulación Numérica del Intercambio de Masas de Agua a través del Estrecho de Gibraltar, Cádiz (2000) 223–239. 
  23. [23] J.B. Schijf and J.C. Schonfeld, Theoretical considerations on the motion of salt and fresh water, in Proc. of the Minn. Int. Hydraulics Conv. Joint meeting IAHR and Hyd. Div. ASCE., Sept. 1953 (1953) 321–333. 
  24. [24] J.J. Stoker, Water Waves. Interscience, New York (1957). Zbl0078.40805
  25. [25] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag, Berlin (1997). Zbl0888.76001MR1474503
  26. [26] M.E. Vázquez-Cendón, Estudio de Esquemas Descentrados para su Aplicación a las leyes de Conservación Hiperbólicas con Términos Fuente. PhD thesis, Universidad de Santiago de Compostela (1994). 
  27. [27] M.E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comp. Physics 148 (1999) 497–526. Zbl0931.76055

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