Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

Marquina, Antonio

doi:10.1016/j.jcp.2009.12.014

Cited by 16 publications

(30 citation statements)

References 33 publications

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“…Then equation (15) becomes the relativistic heat equation (14). The relativistic heat equation has been widely studied from the analytical point of view [1,2,3,4,18,22] and more recently a numerical approach has been presented in [18]. In the later the author proposes a conservative numerical scheme for the approximation of a specific class of Fokker-Planck equations in which the relativistic heat equation is a model problem.…”

Section: Relativistic Heat Equationmentioning

confidence: 99%

“…Many complex physical systems like plasmas, porous media, transport in statistical mechanics, geometric flows among others are described by a class of (anisotropic) diffusion equations that model transport by diffusion of a physical magnitude in a continuum medium ( [1,2,3,4,7,17,18,19,22,23]). Fokker-Planck formulation represents these models whose solutions contain diffusion fronts propagating with finite speed.…”

Section: Introductionmentioning

confidence: 99%

“…The solution to this model for a delta initial signal is a Gaussian distribution defined over all R n for any time t. The main drawback of this model relies on the fact that transport of heat conduction occurs at infinite speed making the model not appropriate for the description of many dissipative processes in thermodynamics. This issue has been widely discussed in the literature and different attempts to overcome it have been presented ( [6,12,13,18,22]). …”

Section: Introductionmentioning

confidence: 99%

“…This transport model allows the formation of diffusion fronts propagating with finite speed under certain conditions on function g and the solution for a compact supported initial data is confined to a bounded region in R n at any time t > 0 ( [1,2,4,5,18]). In our approach we analyze a particular class of limited diffusion FokkerPlanck equations of type (1).…”

Section: Introductionmentioning

confidence: 99%

“…We perform a set of numerical experiments and compare the solution for the three model problems for the limited diffusion equations and their corresponding Hamilton-Jacobi equations. For the numerical approximation of the solution of the limited diffusion Fokker-Planck equations we use the conservative numerical scheme presented in [18]. For the numerical approximation of the new class of Hamilton-Jacobi equations we propose a suitable finite difference numerical scheme to approximate the viscosity solution with shocks arising in the models.…”

Section: Introductionmentioning

confidence: 99%

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

Serna¹,

Marquina²

2013

Applied Numerical Mathematics

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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.

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Section: Relativistic Heat Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Serna¹,

Marquina²

2013

Applied Numerical Mathematics

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Pattern formation in a flux limited reaction–diffusion equation of porous media type

et al. 2016

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Abstract. A nonlinear PDE featuring flux limitation effects together with those of the porous media equation (nonlinear Fokker-Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction-diffusion equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed.

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A convergent numerical scheme for integrodifferential kinetic models of angiogenesis

Bonilla

Carpio

Carretero

et al. 2018

Journal of Computational Physics

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We study a robust finite difference scheme for integrodifferential kinetic systems of Fokker-Planck type modeling tumor driven blood vessel growth. The scheme is of order one and enjoys positivity features. We analyze stability and convergence properties, and show that soliton-like asymptotic solutions are correctly captured. We also find good agreement with the solution of the original stochastic model from which the deterministic kinetic equations are derived working with ensemble averages. A numerical study clarifies the influence of velocity cutoffs on the solutions for exponentially decaying data.

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Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

Cited by 16 publications

References 33 publications

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

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

Pattern formation in a flux limited reaction–diffusion equation of porous media type

A convergent numerical scheme for integrodifferential kinetic models of angiogenesis

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