Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations

Hong Lu; Ji Li; Mingji Zhang

doi:10.3934/dcdsb.2020065

Article Contents

2020, Volume 25, Issue 9: 3357-3371. Doi: 10.3934/dcdsb.2020065

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Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations

1.
School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China
2.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
3.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
4.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

^* Corresponding author: Mingji Zhang
^* Corresponding author: Mingji Zhang

Received: April 30, 2019

Revised: October 31, 2019

Early access: April 2020

Published: September 2020

This work was supported the NSF of China (No. 11601278) and MPS Simons Foundation of USA (No. 628308)

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Abstract

In this paper, we consider the numerical approximation of the space and time fractional Bloch-Torrey equations. A fully discrete spectral scheme based on a finite difference method in the time direction and a Galerkin-Legendre spectral method in the space direction is developed. In order to reduce the amount of computation, an alternating direction implicit (ADI) spectral scheme is proposed. Then the stability and convergence analysis are rigorously established. Finally, numerical results are presented to support our theoretical analysis.

Keywords:

Space and time fractional Bloch-Torrey equations,
Legendre spectral method,
alternating direction implicit method,
stability and convergence.

Mathematics Subject Classification: Primary: 26A33, 65M70; Secondary: 65M12.

Citation:

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Figure 1. The left graph with $ \alpha = 0.3,\ \beta = 0.6 $ while the right one with $ \alpha = 0.8,\ \beta = 0.75 $ for Example 6.1

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Figure 2. $ \alpha = 0.5,\ \beta = 0.8 $ for Example 6.2

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Table 1. $ L^2 $ errors and convergence rates for Example 6.1

$ \tau $	$ \alpha=0.3\ \beta=0.6 $	Con. rate	$ \alpha=0.8\ \beta=0.75 $	Con. rate
1/10	1.4361e-004	2.1596	1.5512e-004	2.0231
1/20	3.2143e-005	2.0832	3.8163e-005	2.0046
1/40	7.5856e-006	2.0480	9.5104e-006	1.9942
1/80	1.8343e-006	1.9814	2.3872e-006	1.9743
1/160	4.6451e-007	1.7299	6.0791e-007	1.8456
1/320	1.4003e-007	-	1.6914e-007	-

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Table 2. $ L^2 $ errors and convergence rates for Example 6.2

$ \tau $	$ \alpha=0.3\ \beta=0.6 $	Con. rate	$ \alpha=0.8\ \beta=0.75 $	Con. rate
1/10	1.7946e-003	2.1361	1.6438e-003	2.0271
1/20	4.0827e-004	2.0638	4.0329e-004	2.0098
1/40	9.7653e-005	2.0361	1.0014e-005	1.9949
1/80	2.3810e-005	2.0142	2.5123e-005	1.9935
1/160	5.8943e-006	1.9872	6.3092e-006	1.9763
1/320	1.4867e-006	-	1.6034e-006	-

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