Numerical methods for one-dimensional aggregation equations
Abstract
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.
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num1D_corr_HAL.pdf (3.95 Mo)
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num1D_v5.pdf (4.88 Mo)
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Origin | Files produced by the author(s) |
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Origin | Files produced by the author(s) |
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