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A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. (English) Zbl 1133.65308

Summary: We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws

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