Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis

International audienceWe introduce a new class of finite difference schemes for approximating the solutions to an initial-boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to non constant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. A special care is needed to deal with boundary conditions, to avoid harmful loss of mass. Convergence results are proven for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior

Numerical methods for one-dimensional aggregation equations

We focus in this work on the numerical discretization of the one dimensional aggregation equation \pa_t\rho + \pa_x (v\rho)=0, $v=a(W'*\rho)$, in the attractive case. Finite time blow up of smooth initial data occurs for potential $W$ having a Lipschitz singularity at the origin. A numerical discretization is proposed for which the convergence towards duality solutions of the aggregation equation is proved. It relies on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$. Moreover, using the same idea, we propose an asymptotic preserving scheme for a kinetic system in hyperbolic scaling converging towards the aggregation equation in hydrodynamical limit. Finally numerical simulations are provided to illustrate the results

A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge