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On a hybrid finite-volume-particle method. (English) Zbl 1077.65091

Summary: We present a hybrid finite-volume-particle numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the one- and two-dimensional Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. This paper is an extension of our previous work [cf. A. Chertock, A. Kurganov and G. Petrova, Finite-volume-particle method for models of transport of pollutant in shallow water, J. Sci. Comput. (to appear)], where the one-dimensional finite-volume-particle method has been proposed.
The core idea behind the finite-volume-particle method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilized at the right place. A special attention is given to the recovery of the point values of the numerical solution from its particle distribution. The reconstruction is obtained using a dual equation for the pollutant concentration. This results in a significantly enhanced resolution of the computed solution and also makes it much easier to extend the finite-volume-particle method to the two-dimensional case.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
76M12 Finite volume methods applied to problems in fluid mechanics
76M28 Particle methods and lattice-gas methods
35Q35 PDEs in connection with fluid mechanics
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References:

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