The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

Author:

Stern Robin,Effenberger Frederic,Fichtner Horst1,Schäfer Tobias2

Affiliation:

1. Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780, Bochum, Germany

2. Department of Mathematics, College of Staten Island, Cuny, NY, USA

Abstract

Abstract The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Analysis

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