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IMA Journal of Applied Mathematics Advance Access originally published online on January 22, 2009
IMA Journal of Applied Mathematics 2009 74(2):201-229; doi:10.1093/imamat/hxn044
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Numerical simulation for the 3D seepage flow with fractional derivatives in porous media

Q. Liu

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

F. Liu{dagger}

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia and School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China

I. Turner and V. Anh

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia

{dagger} Email: f.liu{at}qut.edu.au or fwliu{at}xmu.edu.cn

Received on September 17, 2007; Revision received December 2, 2008. Accepted on December 8, 2008

In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSF-UM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.

Keywords: seepage flow; fractional derivative; fractional alternating direction implicit scheme; modified Douglas scheme; stability and convergence; Richardson extrapolation.


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