Finite difference approximations for a fractional advection diffusion problem
Finite difference approximations for a fractional advection diffusion problem
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DOI：
10.1016/j.jcp.2009.02.011
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发表时间：
2009-06
期刊：
影响因子：
--
通讯作者：
E. Sousa
中科院分区：
文献类型：
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作者：
E. Sousa
近年来，许多研究者对传统的平流扩散方程在许多物理情况下的应用提出了质疑，并提出了替代的扩散模型。分数阶空间导数用于模拟异常扩散或分散，其中粒子羽流以与经典布朗运动模型不一致的速率扩散。当分数阶导数取代扩散或弥散模型中的二阶导数时，它会导致增强扩散，也称为超扩散。考虑一维对流扩散模型，其中通常的二阶导数被α阶分数阶导数取代，1<α <$2。我们推导出显式有限差分格式，这些格式可以视为文献中已有的对流扩散方程格式的推广。我们提出了该计划的精度顺序，为了显示其收敛性，我们证明了他们在一定的条件下是稳定的。最后，我们提出了一个测试问题。
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 α, with 1<α⩽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.