The quadratic approximation of Bernoulli relation for shallow water model
Résumé
The aim of this paper is to propose a new approach to derive a well-balanced Godunov type scheme to approximate the weak solutions of the shallow water equation. here, we propose an extension of the Godunov-type scheme by adopting a new linearization, which relies on a quadratic formulation, of the Bernoulli relation. This new linearization makes it possible to mimic more precisely the exact behavior of stationary solutions. The choice of this new quadratic formulation, allows us to define, without additional correction, a discretization always well defined whatever the Froude number. We establish that the Godunov type finite volume scheme resulting from our developments verifies the essential properties, namely the preservation of the positivity of the water height, the capture of all stationary solutions for a Froude number different to one and finally an entropy inequality defined relatively to perturbation in O(∆x). Several numerical experiments are using to illustrate the relevance of the designed scheme.
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