A strongly degenerate parabolic aggregation equation
Abstract
This paper is concerned with a strongly degenerate convection-diffusion equation in one space dimension whose convective flux involves a non-linear function of the total mass to one side of the given position. This equation can be understood as a model of aggregation of the individuals of a population with the solution representing their local density. The aggregation mechanism is balanced by a degenerate diffusion term accounting for dispersal. In the strongly degenerate case, solutions of the non-local problem are usually discontinuous and need to be defined as weak solutions satisfying an entropy condition. A finite difference scheme for the non-local problem is formulated and its convergence to the unique entropy solution is proved. The scheme emerges from taking divided differences of a monotone scheme for the local PDE for the primitive. Numerical examples illustrate the behaviour of entropy solutions of the non-local problem, in particular the aggregation phenomenon.
Cite
@article{arxiv.1007.1470,
title = {A strongly degenerate parabolic aggregation equation},
author = {Fernando Betancourt and Raimund Bürger and Kenneth H. Karlsen},
journal= {arXiv preprint arXiv:1007.1470},
year = {2010}
}
Comments
30 pages, 8 figures