Numerical approximations for a nonlocal evolution equation

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 solutio...

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Autor principal:	Rossi, Julio Daniel
Publicado:	2011
Materias:	Neumann boundary conditions Nonlocal diffusion Numerical approximations P-Laplacian Sandpiles Neumann boundary condition Laplace transforms Partial differential equations Boundary conditions
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00361429_v49_n5_p2103_PerezLlanos http://hdl.handle.net/20.500.12110/paper_00361429_v49_n5_p2103_PerezLlanos
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00361429_v49_n5_p2103_PerezLlanos
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spelling	paper:paper_00361429_v49_n5_p2103_PerezLlanos2023-06-08T15:01:59Z Numerical approximations for a nonlocal evolution equation Rossi, Julio Daniel 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 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. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2011 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00361429_v49_n5_p2103_PerezLlanos http://hdl.handle.net/20.500.12110/paper_00361429_v49_n5_p2103_PerezLlanos
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	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
spellingShingle	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 Rossi, Julio Daniel Numerical approximations for a nonlocal evolution equation
topic_facet	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
description	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.
author	Rossi, Julio Daniel
author_facet	Rossi, Julio Daniel
author_sort	Rossi, Julio Daniel
title	Numerical approximations for a nonlocal evolution equation
title_short	Numerical approximations for a nonlocal evolution equation
title_full	Numerical approximations for a nonlocal evolution equation
title_fullStr	Numerical approximations for a nonlocal evolution equation
title_full_unstemmed	Numerical approximations for a nonlocal evolution equation
title_sort	numerical approximations for a nonlocal evolution equation
publishDate	2011
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00361429_v49_n5_p2103_PerezLlanos http://hdl.handle.net/20.500.12110/paper_00361429_v49_n5_p2103_PerezLlanos
work_keys_str_mv	AT rossijuliodaniel numericalapproximationsforanonlocalevolutionequation
_version_	1768546059177426944

Numerical approximations for a nonlocal evolution equation

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