Fronts propagating with signal dependent speed in limited diffusion and related Hamilton-Jacobi formulations

Susana Serna, Antonio Marquina

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1 Citation (Scopus)

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. © 2012 IMACS.
Original languageEnglish
Pages (from-to)48-62
JournalApplied Numerical Mathematics
Volume73
DOIs
Publication statusPublished - 1 Jan 2013

Keywords

  • Hamilton-Jacobi equations
  • Limited diffusion equations
  • Numerical approximation
  • Viscosity solutions with shocks

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