2,830 research outputs found
On an explicit finite difference method for fractional diffusion equations
A numerical method to solve the fractional diffusion equation, which could
also be easily extended to many other fractional dynamics equations, is
considered. These fractional equations have been proposed in order to describe
anomalous transport characterized by non-Markovian kinetics and the breakdown
of Fick's law. In this paper we combine the forward time centered space (FTCS)
method, well known for the numerical integration of ordinary diffusion
equations, with the Grunwald-Letnikov definition of the fractional derivative
operator to obtain an explicit fractional FTCS scheme for solving the
fractional diffusion equation. The resulting method is amenable to a stability
analysis a la von Neumann. We show that the analytical stability bounds are in
excellent agreement with numerical tests. Comparison between exact analytical
solutions and numerical predictions are made.Comment: 22 pages, 6 figure
An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator
Boundary Conditions for Fractional Diffusion
This paper derives physically meaningful boundary conditions for fractional
diffusion equations, using a mass balance approach. Numerical solutions are
presented, and theoretical properties are reviewed, including well-posedness
and steady state solutions. Absorbing and reflecting boundary conditions are
considered, and illustrated through several examples. Reflecting boundary
conditions involve fractional derivatives. The Caputo fractional derivative is
shown to be unsuitable for modeling fractional diffusion, since the resulting
boundary value problem is not positivity preserving
Finite difference approximations for a fractional advection diffusion 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
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