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MSC: 35L65, 65M12; International audience; In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where the main difficulty arises in the treatment of the discontinuity occurring in the flux function. In particular, we compare a Roe-type scheme to the well-established Lax-Friedrichs method and provide a numerical study highlighting the benefits of the Roe discretisation. Besides, we also prove the L1-Lipschitz continuous dependence on the initial datum, ensuring the uniqueness of the solution.

International audience; We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax-Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.

International audience; We consider the initial boundary value problem (IBVP) for a non-local scalar conservation laws in one space dimension. The non-local operator in the flux function is not a mere convolution product, but it is assumed to be aware of boundaries. Introducing an adapted Lax-Friedrichs algorithm, we provide various estimates on the approximate solutions that allow to prove the existence of solutions to the original IBVP. The uniqueness follows from the Lipschitz continuous dependence on initial and boundary data, which is proved exploiting results available for the local IBVP.

International audience; We prove the stability of entropy weak solutions of a class of scalar conservation laws with non-local flux arising in traffic modelling. We obtain an estimate of the dependence of the solution with respect to the kernel function, the speed and the initial datum. Stability is obtained from the entropy condition through doubling of variable technique. We finally provide some numerical simulations illustrating the dependencies above for some cost functionals derived from traffic flow applications.

International audience; The well posedness for a class of non local systems of conservation laws in a bounded domain is proved and various stability estimates are provided. This construction is motivated by the modelling of crowd dynamics, which also leads to define a non local operator adapted to the presence of a boundary. Numerical integrations show that the resulting model provides qualitatively reasonable solutions.