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SIAM J. SCI. COMPUT. c 2009 Society for Industrial and Applied Mathematics
Vol. 31, No. 3, pp. 16031627
TWO-LAYER SHALLOW WATER SYSTEM: A RELAXATION
R´EMI ABGRALL AND SMADAR KARNI
Abstract. The two-layer shallow water system is an averaged flow model. It forms a noncon-
servative system which is only conditionally hyperbolic. The coupling between the layers, due to the
hydrostatic pressure assumption, does not provide explicit access to the system eigenstructure, which
is inconvenient for Riemann solution based numerical schemes. We consider a relaxation approach
which offers greater decoupling and accessible eigenstructure. The stability of the model is discussed.
Numerical results are shown for unsteady flows as well as for smooth and nonsmooth steady flows.
Key words. hyperbolic conservation laws, balance laws, shallow water equations, relaxation
AMS subject classifications. 76M12, 35L65, 76N15, 76B70
1.1. Preliminaries. The shallow water equations are commonly used to de-
scribe flows that are nearly horizontal. They can be obtained from the Euler equations
by vertical averaging across the layer depth. The layers are distinguished by different