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Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution

The authors are grateful to the National Natural Science Foundation of PR China (Grant Nos. 11801332, 11571002, and 11971276)

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  • In this work, the time fractional KdV equation with Caputo time derivative of order $ \alpha \in (0,1) $ is considered. The solution of this problem has a weak singularity near the initial time $ t = 0 $. A fully discrete discontinuous Galerkin (DG) method combining the well-known L1 discretisation in time and DG method in space is proposed to approximate the time fractional KdV equation. The unconditional stability result and O$ (N^{-\min \{r\alpha,2-\alpha\}}+h^{k+1}) $ convergence result for $ P^k \; (k\geq 2) $ polynomials are obtained. Finally, numerical experiments are presented to illustrate the efficiency and the high order accuracy of the proposed scheme.

    Mathematics Subject Classification: Primary: 35R11, 65M60; Secondary: 65M12.

    Citation:

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  • Figure 1.  The numerical solution for Example 4.1 with $ \alpha = 0.4 $

    Table 1.  $ L^\infty(L^2) $ errors and orders of convergence on temporal direction for Example 4.1 with $ r = (2-\alpha)/\alpha $

    N = 32N = 64N = 128N = 256 $ N = 512 $N = 1024
    $ \alpha = 0.4 $3.0496E-21.1110E-23.9235E-31.3578E-34.6379E-41.5729E-4
    1.45671.50161.53071.54981.5600
    $ \alpha = 0.6 $3.8341E-21.5127E-25.8825E-32.2665E-38.6831E-43.3157E-4
    1.34171.36261.37591.38421.3888
    $ \alpha = 0.8 $5.9953E-22.6607E-21.1728E-25.1485E-32.2540E-39.8512E-4
    1.17201.18171.18781.19161.1941
     | Show Table
    DownLoad: CSV

    Table 2.  Errors and orders of convergence on space direction for Example 4.1 with $ \alpha = 0.4 $

    Polynomial M $ \|u-u_h\|_{L^2} $ Order $ \|u-u_h\|_{L^{\infty}} $ Order
    $ P^2 $ 5 5.3831E-01 - 3.2328E-01 -
    10 7.8579E-02 2.7762 4.7729E-02 2.7598
    20 9.9319E-03 2.9840 6.2124E-03 2.9416
    40 1.1426E-04 3.1196 7.5845E-04 3.0340
    $ P^3 $ 5 1.7236E-02 - 1.3819E-02 -
    10 1.1399E-03 3.9184 8.7589E-04 3.9798
    15 2.2712E-04 3.9406 1.7695E-04 3.9667
    20 7.2979E-05 3.9418 6.1408E-04 3.9070
     | Show Table
    DownLoad: CSV

    Table 3.  $ L^\infty(L^2) $ errors and orders of convergence on temporal direction for Example 4.2 with $ r = (2-\alpha)/\alpha $

    N = 32N = 64N = 128N = 256 $ N = 512 $N = 1024
    $ \alpha = 0.4 $2.6605E-29.8042E-33.4860E-31.2119E-34.1549E-41.4179E-4
    1.44021.49181.52431.54441.5510
    $ \alpha = 0.6 $3.0086E-21.2002E-24.6980E-31.8177E-36.9850E-42.6770E-4
    1.32581.35311.36991.37971.3836
    $ \alpha = 0.8 $4.1374E-21.8226E-27.9818E-33.4836E-31.5178E-36.6104E-4
    1.18271.19121.19611.19851.1992
     | Show Table
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    Table 4.  Errors and orders of convergence on space direction for Example 4.2 with $ \alpha = 0.4 $

    Polynomial M $ \|u-u_h\|_{L^2} $ Order $ \|u-u_h\|_{L^{\infty}} $ Order
    $ P^2 $ 5 3.8931E-01 - 2.3483E-01 -
    10 5.6563E-02 2.7829 3.4418E-02 2.7704
    20 7.1139E-03 2.9911 4.4696E-03 2.9449
    40 7.7979E-04 3.1894 5.4210E-04 3.0435
    $ P^3 $ 5 1.2812E-02 - 1.0564E-02 -
    10 8.3809E-03 3.9342 6.7270E-04 3.9731
    15 1.6615E-04 3.9552 1.2940E-04 4.0071
    20 5.3162E-05 3.9564 4.3749E-05 3.9578
     | Show Table
    DownLoad: CSV

    Table 5.  $ L^2 $ errors and orders of convergence on temporal direction for Example 4.3 with $ r = (2-\alpha)/\alpha $

    N = 64N = 128N = 256N = 512N = 1024
    $ \alpha = 0.4 $2.4959E-38.4989E-42.8741E-49.6635E-53.2367E-5
    1.55421.56411.57201.5784
    $ \alpha = 0.6 $6.0125E-32.3383E-38.9915E-43.4367E-41.3092E-4
    1.36241.37881.38751.3923
    $ \alpha = 0.8 $1.0359E-24.6808E-32.0899E-31.2645E-44.0879E-4
    1.14601.16331.17361.1803
     | Show Table
    DownLoad: CSV

    Table 6.  $ L^2 $ errors and orders of convergence on temporal direction for Example 4.3 with $ r = 2(2-\alpha)/\alpha $

    N = 64N = 128N = 256N = 512N = 1024
    $ \alpha = 0.4 $6.0380E-32.2285E-37.2999E-42.4866E-48.4023E-5
    1.51101.53711.55361.5653
    $ \alpha = 0.6 $1.1214E-24.4769E-31.7480E-36.7438E-42.5839E-4
    1.32481.35671.37411.3839
    $ \alpha = 0.8 $1.5602E-27.0291E-33.1157E-31.3694E-35.9926E-4
    1.15031.17371.18591.1923
     | Show Table
    DownLoad: CSV
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