Fast ADI method for high dimensional fractional diffusion equations in conservative form with preconditioned strategy

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• 作者：Lot-Kei Chou ^{yb67435@umac.mo} ; Siu-Long Lei ; ^{sllei@umac.mo}
• 关键词：High dimensional two-sided fractional diffusion equation ; Alternating direction implicit method ; Approximate inverse preconditioner ; Krylov subspace method ; Superlinear convergence ; Fast Fourier transform
• 刊名：Computers & Mathematics with Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：73
• 期：3
• 页码：385-403
• 全文大小：506 K
• 卷排序：73

文摘

In this paper, high dimensional two-sided space fractional diffusion equations, derived from the fractional Fick’s law, and with monotonic variable diffusion coefficients, are solved by alternating direction implicit method. Each linear system corresponding to each spatial direction thus resulted is solved by Krylov subspace method. The method is accelerated by applying an approximate inverse preconditioner, where under certain conditions we showed that the normalized preconditioned matrix is equal to a sum of identity matrix, a matrix with small norm, and a matrix with low rank, such that the preconditioned Krylov subspace method converges superlinearly. We also briefly present some fast algorithms whose computational cost for solving the linear systems is O(nlogn)O(nlogn), where nn is the matrix size. The results are illustrated by some numerical examples.