Fronts propagating with signal dependent speed in limited diffusion and related Hamilton-Jacobi formulations
Serna, Susana (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Marquina, Antonio (Universitat de València. Departament de Matemàtiques)

Date:	2013
Abstract:	We consider a class of limited diffusion equations and explore the formation of diffusion fronts as the result of a combination of diffusive and hyperbolic transport. We analyze a new class of Hamilton-Jacobi equations arising from the convective part of general Fokker-Planck equations ruled by a non-negative diffusion coefficient that depends on the unknown and on the gradient of the unknown. We explore the main features of the solution of the Hamilton-Jacobi equations that contain shocks and propose a suitable numerical scheme that approximates the solution in a consistent way with respect to the solution of the associated Fokker-Planck equation. We analyze three model problems covering different scenarios. One is the relativistic heat equation model where the speed of propagation of fronts is constant. A second one is a standard porous media model where the speed of propagation of fronts is a function of the density, is unbounded and can exceed any fixed value. We propose a third one which is a porous media model whose speed of propagating fronts depends on the density media and is limited. The three model problems satisfy a general Darcy law. We perform a set of numerical experiments under different piecewise smooth initial data with compact support and compare the behavior of the three different model problems.
Grants:	Ministerio de Ciencia e Innovación MTM2011-28043 Ministerio de Ciencia e Innovación MTM2011-26995-C02-01 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-345
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Limited diffusion equations ; Hamilton-Jacobi equations ; Viscosity solutions with shocks ; Numerical approximation
Published in:	Applied Numerical Mathematics, Vol. 73 (2013) , p. 48-62, ISSN 0168-9274

DOI: 10.1016/j.apnum.2012.07.006

Postprint 26 p, 1.5 MB

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