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21-06-2019 | Original Paper | Issue 1/2020

Numerical Algorithms 1/2020

Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations

Journal:
Numerical Algorithms > Issue 1/2020
Authors:
Yun-Chi Huang, Siu-Long Lei
Important notes
This research was partially supported by the research grants MYRG2016-00202-FST, MYRG2018-00025-FST from University of Macau and 048/2017/A from Macao Science and Technology Development Fund (FDCT).

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Abstract

Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires \(\mathcal {O}\left ({NM}_{1}M_{2}\left ({M_{1}^{2}}+{M_{2}^{2}}+NM_{1}M_{2}\right )\right )\) complexity and \(\mathcal {O}\left ({N{M_{1}^{2}}{M_{2}^{2}}}\right )\) storage, where N is the number of temporal unknown and M1, M2 are the numbers of spatial unknown in x, y directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage \(\mathcal {O}\left ({NM}_{1}M_{2}\right )\) are developed. The complexity for the fast solver via time-marching is \(\mathcal {O}\left ({NM}_{1}M_{2}\left (N+\log \left (M_{1}M_{2}\right )\right )\right )\) and the one via divide-and-conquer technique is \(\mathcal {O}\left ({NM}_{1}M_{2}\left (\log ^{2} N+ \log \left (M_{1}M_{2}\right )\right )\right )\). It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.

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