Commun. Comput. Phys., 13 (2013), pp. 502-525. |
Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential Aderito Araujo ^{1}, Amal K. Das ^{2}, Cidalia Neves ^{3}, Ercilia Sousa ^{1*} 1 CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal.2 Department of Physics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. 3 CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal; and ISCAC, Polytechnic Institute of Coimbra, 3040-316 Coimbra, Portugal. Received 28 July 2011; Accepted (in revised version) 1 March 2012 Available online 17 July 2012 doi:10.4208/cicp.280711.010312a Abstract Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented. AMS subject classifications: 35L15, 60K40, 60J65, 65R10, 65M06, 65M12PACS: 02.60.Lj, 05.40.Jc Key words: Numerical methods, Laplace transform, telegraph equation, periodic potential, non-Fickian diffusion. *Corresponding author. Email: alma@mat.uc.pt (A. Araujo), akdas@dal.ca (A. K. Das), cneves@iscac.pt (C. Neves), ecs@mat.uc.pt (E. Sousa) |