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Abstract:

In this paper we study numerical approximations of continuous solutions to the nonlocal p-Laplacian type diffusion equation, ut(t, x) = ∫Ω J(x - y)|u(t, y) - u(t, x)|p-2(u(t, y) - u(t, x)) dy. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties of the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity. Next, we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties. In addition, we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile. Finally, we present some numerical experiments that illustrate our results. © by SIAM. Unauthorized reproduction of this article is prohibited. © 2011 Society for Industrial and Applied Mathematics.

Registro:

Documento: Artículo
Título:Numerical approximations for a nonlocal evolution equation
Autor:Pérez-Llanos, M.; Rossi, J.D.
Filiación:Departamento de Matemática, Universidad Autónoma de Madrid, Campus Cantoblanco, 28049 Madrid, Spain
Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, 03080 Alicante, Spain
Departamento de Matemática, FCEyN UBA (1428), Buenos Aires, Argentina
Palabras clave:Neumann boundary conditions; Nonlocal diffusion; Numerical approximations; P-Laplacian; Sandpiles; Neumann boundary condition; Nonlocal diffusion; Numerical approximations; P-Laplacian; Sandpiles; Laplace transforms; Partial differential equations; Boundary conditions
Año:2011
Volumen:49
Número:5
Página de inicio:2103
Página de fin:2123
DOI: http://dx.doi.org/10.1137/110823559
Título revista:SIAM Journal on Numerical Analysis
Título revista abreviado:SIAM J Numer Anal
ISSN:00361429
CODEN:SJNAA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00361429_v49_n5_p2103_PerezLlanos

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Citas:

---------- APA ----------
Pérez-Llanos, M. & Rossi, J.D. (2011) . Numerical approximations for a nonlocal evolution equation. SIAM Journal on Numerical Analysis, 49(5), 2103-2123.
http://dx.doi.org/10.1137/110823559
---------- CHICAGO ----------
Pérez-Llanos, M., Rossi, J.D. "Numerical approximations for a nonlocal evolution equation" . SIAM Journal on Numerical Analysis 49, no. 5 (2011) : 2103-2123.
http://dx.doi.org/10.1137/110823559
---------- MLA ----------
Pérez-Llanos, M., Rossi, J.D. "Numerical approximations for a nonlocal evolution equation" . SIAM Journal on Numerical Analysis, vol. 49, no. 5, 2011, pp. 2103-2123.
http://dx.doi.org/10.1137/110823559
---------- VANCOUVER ----------
Pérez-Llanos, M., Rossi, J.D. Numerical approximations for a nonlocal evolution equation. SIAM J Numer Anal. 2011;49(5):2103-2123.
http://dx.doi.org/10.1137/110823559