A strongly degenerate parabolic aggregation equation

TL;DR

This paper studies a strongly degenerate aggregation model combining non-linear convection and degenerate diffusion, proving convergence of a finite difference scheme to the entropy solution and illustrating aggregation behavior through numerical examples.

Contribution

It introduces a finite difference scheme for a non-local degenerate PDE and proves its convergence to the unique entropy solution, addressing discontinuities.

Findings

Finite difference scheme converges to the entropy solution.

Numerical examples demonstrate aggregation phenomena.

The scheme effectively handles discontinuous solutions.

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.