Article Contents
Article Contents

# 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)

• 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:

• 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 = 32 N = 64 N = 128 N = 256 $N = 512$ N = 1024 $\alpha = 0.4$ 3.0496E-2 1.1110E-2 3.9235E-3 1.3578E-3 4.6379E-4 1.5729E-4 1.4567 1.5016 1.5307 1.5498 1.5600 $\alpha = 0.6$ 3.8341E-2 1.5127E-2 5.8825E-3 2.2665E-3 8.6831E-4 3.3157E-4 1.3417 1.3626 1.3759 1.3842 1.3888 $\alpha = 0.8$ 5.9953E-2 2.6607E-2 1.1728E-2 5.1485E-3 2.2540E-3 9.8512E-4 1.1720 1.1817 1.1878 1.1916 1.1941

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

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

 N = 32 N = 64 N = 128 N = 256 $N = 512$ N = 1024 $\alpha = 0.4$ 2.6605E-2 9.8042E-3 3.4860E-3 1.2119E-3 4.1549E-4 1.4179E-4 1.4402 1.4918 1.5243 1.5444 1.5510 $\alpha = 0.6$ 3.0086E-2 1.2002E-2 4.6980E-3 1.8177E-3 6.9850E-4 2.6770E-4 1.3258 1.3531 1.3699 1.3797 1.3836 $\alpha = 0.8$ 4.1374E-2 1.8226E-2 7.9818E-3 3.4836E-3 1.5178E-3 6.6104E-4 1.1827 1.1912 1.1961 1.1985 1.1992

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

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

 N = 64 N = 128 N = 256 N = 512 N = 1024 $\alpha = 0.4$ 2.4959E-3 8.4989E-4 2.8741E-4 9.6635E-5 3.2367E-5 1.5542 1.5641 1.5720 1.5784 $\alpha = 0.6$ 6.0125E-3 2.3383E-3 8.9915E-4 3.4367E-4 1.3092E-4 1.3624 1.3788 1.3875 1.3923 $\alpha = 0.8$ 1.0359E-2 4.6808E-3 2.0899E-3 1.2645E-4 4.0879E-4 1.1460 1.1633 1.1736 1.1803

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

 N = 64 N = 128 N = 256 N = 512 N = 1024 $\alpha = 0.4$ 6.0380E-3 2.2285E-3 7.2999E-4 2.4866E-4 8.4023E-5 1.5110 1.5371 1.5536 1.5653 $\alpha = 0.6$ 1.1214E-2 4.4769E-3 1.7480E-3 6.7438E-4 2.5839E-4 1.3248 1.3567 1.3741 1.3839 $\alpha = 0.8$ 1.5602E-2 7.0291E-3 3.1157E-3 1.3694E-3 5.9926E-4 1.1503 1.1737 1.1859 1.1923
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