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Abstract: This paper is concerned with a strongly degenerate convection-diffusionequation in one space dimension whose convective flux involves a non-linearfunction of the total mass to one side of the given position. This equation canbe understood as a model of aggregation of the individuals of a population withthe solution representing their local density. The aggregation mechanism isbalanced by a degenerate diffusion term accounting for dispersal. In thestrongly degenerate case, solutions of the non-local problem are usuallydiscontinuous and need to be defined as weak solutions satisfying an entropycondition. A finite difference scheme for the non-local problem is formulatedand its convergence to the unique entropy solution is proved. The schemeemerges from taking divided differences of a monotone scheme for the local PDEfor the primitive. Numerical examples illustrate the behaviour of entropysolutions of the non-local problem, in particular the aggregation phenomenon.

Author: Fernando Betancourt, Raimund Bürger, Kenneth H. Karlsen

Source: https://arxiv.org/


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