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Local meshless differential quadrature collocation method for time-fractional PDEs

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  • This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of $ \alpha \in (0,1) $ and $ \alpha\in(1,2) $. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

    Mathematics Subject Classification: Primary: 65M99, 35K55, 35K57; Secondary: 35R11.


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  • Figure 1.  Schematics of central local supported domain in 2D geometry for $ n_i = 5 $

    Figure 2.  Schematic of upwind local supported domain in 1D (row 1) and 2D (row 2) geometry for $ n_i = 3 $ and $ n_i = 5 $ respectively

    Figure 3.  Numerical solutions of the GMM for Test Problem 2

    Figure 4.  Numerical solutions of the LMM with out upwind technique (left), with upwind technique (right) for Test Problem 2

    Figure 5.  Numerical solutions of the LMM with upwind technique for Test Problem 2

    Figure 6.  Computational domain and absolute error for Test Problem 3

    Figure 7.  Computational domain and absolute error for Test Problem 3

    Figure 8.  Computational domain and absolute error for Test Problem 3

    Figure 9.  Computational domain and numerical solution for Test Problem 3

    Figure 10.  Computational domain, approximate and exact solution for Test Problem 3

    Figure 11.  Numerical solution of the GMM for $ Re = 200 $ (left) and $ Re = 300 $ (right) for Test Problem 4

    Figure 12.  Numerical solution of the LMM for $ Re = 100 $ (left) and $ Re = 150 $ (right) for Test Problem 4

    Figure 13.  Results of the LMM using upwind technique for $ Re = 150 $ (left) and $ Re = 1000 $ (right) for Test Problem 4

    Figure 14.  Results of the LMM using upwind technique for $ Re = 10^{10} $ (left) and $ Re = 10^{17} $ (right) for Test Problem 4

    Figure 15.  CPU time comparison of the LMM and the GMM for Test Problem 4

    Table 1.  Comparison of the LMM with different local sub-domain $ n_i $ and the method reported in [8] for Test Problem 1

    Time t=1 t=2 t=2.5 t=3
    Max. abs. error[8] 4.632e-03 5.267e-03 5.569e-03 5.857e-03
    LMM ($ L_{\infty} $)
    $ n_i=3 $ 3.6104e-04 6.3774e-04 7.8362e-04 9.2358e-04
    $ n_i=5 $ 1.6807e-05 2.5918e-05 2.8962e-05 3.3891e-05
    $ n_i=7 $ 7.4546e-06 1.1882e-05 1.3669e-05 1.5306e-05
    $ n_i=9 $ 6.5724e-06 1.0753e-05 1.2321e-05 1.3640e-05
    $ n_i=11 $ 6.4933e-06 1.0749e-05 1.2303e-05 1.3355e-05
     | Show Table
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    Table 2.  $ Ave.L_{abs} $ error norms of the LMM for Test Problem 3

    N 5 10 15 20
    $ \alpha=1.5 $ 2.7911e-03 3.3331e-04 6.3412e-05 1.1145e-05
    $ \alpha=1.8 $ 2.8124e-03 3.5870e-04 8.4685e-05 2.7917e-05
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical results of the LMM and the method reported in [28] for Test Problem 3

    $ \alpha=1.5 $ $ \alpha=1.8 $
    $ \tau $ $ Ave.L_{abs} $ $ Ave.L_{abs} $ [28] $ \tau $ $ Ave.L_{abs} $ $ Ave.L_{abs} $ [28]
    1/10 4.3826e-02 1.2550e-02 1/10 2.9099e-02 2.0496e-02
    1/20 1.2592e-02 6.6277e-03 1/20 9.2610e-03 1.0696e-02
    1/30 6.3054e-03 4.5292e-03 1/30 4.4599e-03 7.2811e-03
    1/40 3.5999e-03 3.4518e-03 1/40 2.3587e-03 5.5407e-03
    1/50 2.1402e-03 2.7951e-03 1/50 1.1993e-03 4.4822e-03
    1/60 1.2811e-03 2.3526e-03 1/60 5.8970e-04 3.7688e-03
    1/70 8.3729e-04 2.0338e-03 1/70 4.7926e-04 3.2548e-03
     | Show Table
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    Table 4.  $ Ave.L_{abs} $ of the LMM for Test Problem 3

    $ \alpha $ Regular nodes Chebyshev nodes
    Explicit CN Implicit Explicit CN Implicit
    1.5 8.0527e-05 2.2050e-04 4.1275e-04 2.4177e-04 3.4777e-04 4.5997e-04
    1.6 8.1357e-05 2.2263e-04 4.1881e-04 2.3932e-04 3.4587e-04 4.5902e-04
    1.7 8.2287e-05 2.2266e-04 4.2312e-04 2.3691e-04 3.4459e-04 4.5928e-04
    1.8 8.3535e-05 2.2109e-04 4.2679e-04 2.3405e-04 3.4367e-04 4.6080e-04
     | Show Table
    DownLoad: CSV
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