A strongly degenerate parabolic aggregation equation

• Main Authors: Betancourt, Fernando, Bürger, Raimund, Karlsen, Kenneth H
• Format: Journal Article
• Language: English
• Published: 08.07.2010
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.1007.1470

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.