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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 Valencia. Departament de Matemàtica Aplicada)

Data: 2013
Resum: 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.
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; recerca ; preprint
Matèria: Limited diffusion equations ; Hamilton-Jacobi equations ; Viscosity solutions with shocks ; Numerical approximation
Publicat a: Applied Numerical Mathematics, Vol. 73 (2013) , p. 48-62, ISSN 0168-9274

DOI: 10.1016/j.apnum.2012.07.006

26 p, 1.5 MB

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