Author(s):
Sousa, Ercília 
Date: 2008
Persistent ID: http://hdl.handle.net/10316/11247
Origin: Estudo Geral - Universidade de Coimbra
Subject(s): Fractional advection diffusion; Finite differences; Stability
Document details
Finite difference approximations for a fractional advection diffusion problem
Description
The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one dimensional advection-diffusion model, where the usual second-order derivative gives place to a fractional derivative of order , with 1 < ≤ 2. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.
The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one dimensional advection-diffusion model, where the usual second-order derivative gives place to a fractional derivative of order , with 1 < ≤ 2. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.
Document Type
Preprint
Language English
Language English
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