A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. (English) Zbl 1226.76008

Kurganov, Alexander; Petrova, Guergana

Commun. Math. Sci. 5, No. 1, 133-160 (2007).

Summary: A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations was first introduced in [the first author and D. Levy, M2AN, Math. Model. Numer. Anal. 36, No. 3, 397-425 (2002; Zbl 1137.65398)]. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but they fail to satisfy both properties simultaneously.
Here, we introduce an improved second-order central-upwind scheme which, unlike its forerunners, is capable of both preserving stationary steady states (lake at rest) and guaranteeing the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness, in a number of one- and two-dimensional examples.

MSC:

76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35L60	First-order nonlinear hyperbolic equations
35Q53	KdV equations (Korteweg-de Vries equations)
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M25	Other numerical methods (fluid mechanics) (MSC2010)

Citations:

Zbl 1137.65398