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

  • * Corresponding author: Mingji Zhang

    * Corresponding author: Mingji Zhang
This work was supported the NSF of China (No. 11601278) and MPS Simons Foundation of USA (No. 628308)
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  • 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.

    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

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

    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 -
     | Show Table
    DownLoad: CSV

    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 -
     | Show Table
    DownLoad: CSV
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