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 -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 -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 -scheme obtained is applied to the two-layer shallow water system.
Mots-clés : Q-schemes, coupled conservation laws, source terms, 1D shallow water equations, two-layer flows, hyperbolic systems
@article{M2AN_2001__35_1_107_0,
author = {Castro, Manuel and Mac{\'\i}as, Jorge and Par\'es, Carlos},
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},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {107--127},
publisher = {EDP-Sciences},
volume = {35},
number = {1},
year = {2001},
mrnumber = {1811983},
zbl = {1094.76046},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_1_107_0/}
}
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: Modélisation mathématique et analyse numérique PY - 2001 SP - 107 EP - 127 VL - 35 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/item/M2AN_2001__35_1_107_0/ LA - en ID - M2AN_2001__35_1_107_0 ER -
%0 Journal Article %A Castro, Manuel %A Macías, Jorge %A Parés, Carlos %T 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 %J ESAIM: Modélisation mathématique et analyse numérique %D 2001 %P 107-127 %V 35 %N 1 %I EDP-Sciences %U http://www.numdam.org/item/M2AN_2001__35_1_107_0/ %G en %F M2AN_2001__35_1_107_0
Castro, Manuel; Macías, Jorge; 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: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 1, pp. 107-127. http://www.numdam.org/item/M2AN_2001__35_1_107_0/
[1] , The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163 (1986) 27-58.
[2] and, Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164 (1986) 27-51. | Zbl
[3] and, Upwind methods for hyperbolic conservation laws with source terms. Computers and Fluids 23 (1994) 1049-1071. | Zbl
[4] and, 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. | Zbl
[5] , and, 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] , and, 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] ,,, and, Une méthode numérique decentrée pour la résolution d'ecoulements diphasiques. C. R. Acad. Sci. Paris, Sér. I 324 (1997) 717-723. | Zbl
[8] , Two-layer Hydraulics Maximal Exchange Flows. Ph.D. thesis, University of Cambridge (1988).
[9] and, 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. | Zbl
[10] and, 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] , and, 1D open channel flow simulation using TVD McCormack scheme. J. Hydraul. Eng. 118 (1992) 1359-1373.
[12] , Atmosphere-Ocean Dynamics, Int. Geophys. Series 30, Springer-Verlag, San Diego (1982) 662 p.
[13] and, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Sc. 118, Springer-Verlag, New York (1996). | MR | Zbl
[14] , On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21 (1984) 1-23. | Zbl
[15] , and, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl
[16] , Time-dependent two-layer hydraulic exchange flows. J. Phys. Oceanogr. 25 (1995) 359-373.
[17] , Fluid Mechanics. Academic Press Inc., San Diego (1990) 638 p. | Zbl
[18] and, Existence theory for the Riemann problem for non-conservative hyperbolic systems. C. R. Acad. Sci. Paris, Sér. I 323 (1996) 347-352. | Zbl
[19] and, Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30 (1999) 1309-1342. | Zbl
[20] , Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357-371. | Zbl
[21] , 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. | Zbl
[22] and, 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] and, 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] , Water Waves. Interscience, New York (1957). | Zbl
[25] , Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag, Berlin (1997). | MR
[26] , 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] , 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. | Zbl
