Geodesic
Numerical analysis for a three interacting species model with nonlocal and cross diffusion
Anaya, Verónica1 ; Bendahmane, Mostafa2 ; Sepúlveda, Mauricio3
1 Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile.
2 Institut Mathematiques de Bordeaux, Universite Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Bordeaux, France.
3 CI2MA & Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 171-192

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In this paper, we consider a reaction-diffusion system describing three interacting species in the food chain structure with nonlocal and cross diffusion. We propose a semi-implicit finite volume scheme for this system, we establish existence and uniqueness of the discrete solution, and it is also showed that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. The convergence proof is based on the use of the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L 1 compactness argument that mimics the compactness lemma due to Kruzhkov. Finally we give some numerical examples.

Reçu le :
DOI : 10.1051/m2an/2014028
Classification : 35K57, 35M10, 35A05
Keywords: Nonlocal and cross diffusion, food chain model, finite volume scheme

Anaya, Verónica 1 ; Bendahmane, Mostafa 2 ; Sepúlveda, Mauricio 3

1 Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile.
2 Institut Mathematiques de Bordeaux, Universite Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Bordeaux, France.
3 CI2MA & Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. 
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     author = {Anaya, Ver\'onica and Bendahmane, Mostafa and Sep\'ulveda, Mauricio},
     title = {Numerical analysis for a three interacting species model with nonlocal and cross diffusion},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {171--192},
     publisher = {EDP-Sciences},
     volume = {49},
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     doi = {10.1051/m2an/2014028},
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     zbl = {1314.65115},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014028/}
}
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Anaya, Verónica; Bendahmane, Mostafa; Sepúlveda, Mauricio. Numerical analysis for a three interacting species model with nonlocal and cross diffusion. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 171-192. doi: 10.1051/m2an/2014028

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