Propagation of Diffusing Pollutant by a Hybrid Eulerian-Lagrangian Method

We present a hybrid numerical method for computing the propagation of a diffusing passive pollutant in shallow water. The flow is modeled by the SaintVenant system of shallow water equations and the pollutant propagation is described by a convection-diffusion equation. In this paper, we extend the hybrid finite-volume-particle (FVP) method, which was originally introduced in [CK04, CKP06] for the model of inviscid pollutant propagation, to the case of a diffusing pollutant. The idea behind the FVP method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finitevolume scheme, while the transport equation for the propagation of passive pollutant is solved by a deterministic particle method. When the pollutant diffuses, the second step of the FVP method has to be modified. We propose a new hybrid Eulerian-Lagrangian method, in which the convection term is treated by the method of characteristics (which is, in the context of scalar transport equations is very similar to the deterministic particle method), while the diffusion is resolved using the fast explicit operator splitting method recently developed in [CKP].