In this paper we present a deterministic numerical approximation of the coalescence or Smoluchowski equation. The proposed numerical scheme conserves the first order momentum and deals with non-uniform grids. The generalization to a multidimensional framework is also described. We validate the scheme considering some classical tests both in one and two dimensions and discuss its behavior when gelation occurs. Our numerical results and code are compared with those already existent in literature.

Starting from discrete event simulations based on sampled data we simulate the interplay between product density and flux. Data-fitting helps to determine the right parameters for clearing functions to close first and second order conservation laws. For the first order case well-known relations from M/M/1-queuing theory can be reproduced and numerically extended to transient behavior. To include more information from the data into the model, a second equation is introduced leading to a second order production model which is close to the Aw-Rascle-Zhang model known from traffic flow. Numerical comparisons show similarities and, in particular, differences of the modeling approaches.

International audienceGrowth of gas bubbles in magmas may be modeled by a system of differential equations that account for the evolution of bubble radius and internal pressure and that are coupled with an advection-diffusion equation defining the gas flux going from magma to bubble. This system of equations is characterized by two relaxation parameters linked to the viscosity of the magma and to the diffusivity of the dissolved gas, respectively. Here, we propose a numerical scheme preserving, by construction, the total mass of water of the system. We also study the asymptotic behavior of the system of equations by letting the relaxation parameters vary from 0 to infinity, and show the numerical convergence of the solutions obtained by means of the general numerical scheme to the simplified asymptotic limits. Finally, we validate and compare our numerical results with those obtained in experiments

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We propose a deterministic method designed for unsteady flows, based on a discretization of the Boltzmann (BGK) equation with local adaptive velocity grids. These grids dynamically adapt in time and space to the variations of the width of the distribution functions. This allows a significant reduction of the memory storage and CPU time, as compared to standard discrete velocity methods, and avoid the delicate problem to construct a priori a sufficient global velocity grid.