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January  2020, 25(1): 321-334. doi: 10.3934/dcdsb.2019185

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

Na An 1, , Chaobao Huang 2,, and  Xijun Yu 3,

1. 

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2. 

School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China
3. 

Labroatory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

* Corresponding author: huangcb@csrc.ac.cn

Received  January 2018 Revised  March 2019 Published  September 2019

Fund Project: 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.

Keywords: Time fractional KdV equation, weak singularity, discontinuous Galerkin method, stability, error estimate.
Mathematics Subject Classification: Primary: 35R11, 65M60; Secondary: 65M12.
Citation: Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
References:
[1]

N. An, C. Huang and X. Yu, Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation, Appl. Math. Comput., 349 (2019), 148-157.  doi: 10.1016/j.amc.2018.12.048.  Google Scholar

[2]

W. Bu and A. Xiao, An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions, Numer. Algor., 81 (2019), 529-545.  doi: 10.1007/s11075-018-0559-2.  Google Scholar

[3]

H. Chen and T. Sun, A Petrov-Galerkin spectral method for the linearized time fractional KdV equation, Int. J. Comput. Math., 95 (2018), 1292-1307.  doi: 10.1080/00207160.2017.1410544.  Google Scholar

[4]

Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp., 77 (2008), 699-730.  doi: 10.1090/S0025-5718-07-02045-5.  Google Scholar

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P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.  Google Scholar

[6]

B. Cockburn and K. Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems, Numer. Math., 130 (2015), 293-314.  doi: 10.1007/s00211-014-0661-x.  Google Scholar

[7]

P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, volume 16 of Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2000.  Google Scholar

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M. Fung, Kdv equation as an euler-poincare equation, Chinese J. Phys., 35 (1997), 789-796.   Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

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R. Hilfer, editor., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

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C. Huang, N. An and X. Yu, A fully discrete direct discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Anal., 97 (2018), 659-675.  doi: 10.1080/00036811.2017.1281407.  Google Scholar

[12]

C. Huang, M. Stynes and N. An, Optimal ${L}^\infty ({L}^2)$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem, BIT. Numer. Math, 58 (2018), 661-690.  doi: 10.1007/s10543-018-0707-z.  Google Scholar

[13]

C. Huang, X. Yu, C. Wang, Z. Li and N. An, A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation, Appl. Math. Comput., 264 (2015), 483-492.  doi: 10.1016/j.amc.2015.04.093.  Google Scholar

[14]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

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Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.  Google Scholar

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S. Momani and A. Yıldı rım, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method, Int. J. Comput. Math., 87 (2010), 1057-1065.  doi: 10.1080/00207160903023581.  Google Scholar

[18]

D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138-1145.  doi: 10.1016/j.camwa.2008.02.015.  Google Scholar

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K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995.  doi: 10.1090/S0025-5718-09-02234-0.  Google Scholar

[20]

K. Mustapha and W. McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal., 32 (2012), 906-925.  doi: 10.1093/imanum/drr027.  Google Scholar

[21]

K. Mustapha, M. Nour and B. Cockburn, Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems, Adv. Comput. Math., 42 (2016), 377-393.  doi: 10.1007/s10444-015-9428-x.  Google Scholar

[22]

I. Podlubny, Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.  Google Scholar

[23]

I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386. Dedicated to the 60th anniversary of Prof. Francesco Mainardi.  Google Scholar

[24]

J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray. Google Scholar

[25]

M. Stynes, E. O'Riordan and J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079.  doi: 10.1137/16M1082329.  Google Scholar

[26]

I. Turner, F. Liu, V. Anh and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089.  Google Scholar

[27]

L. Wei, Y. He, A. Yildirim and S. Kumar, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, ZAMM Z. Angew. Math. Mech., 93 (2013), 14-28.  doi: 10.1002/zamm.201200003.  Google Scholar

[28]

G. H. Weiss, R. Klages, G. Radons and I. M. Sokolov (eds.), Anomalous transport: Foundations and applications [book review of WILEY-VCH Verlag GmbH & Co., Weinheim, 2008], J. Stat. Phys., 135 (2009), 389-391. doi: 10.1007/s10955-009-9713-5.  Google Scholar

[29]

G. B. Witham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[30]

N. Zabusky and M. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

[31]

Q. Zhang, J. Zhang, S. Jiang and Z. Zhang, Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. Comput., 87 (2018), 693-719.  doi: 10.1090/mcom/3229.  Google Scholar

show all references

References:
[1]

N. An, C. Huang and X. Yu, Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation, Appl. Math. Comput., 349 (2019), 148-157.  doi: 10.1016/j.amc.2018.12.048.  Google Scholar

[2]

W. Bu and A. Xiao, An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions, Numer. Algor., 81 (2019), 529-545.  doi: 10.1007/s11075-018-0559-2.  Google Scholar

[3]

H. Chen and T. Sun, A Petrov-Galerkin spectral method for the linearized time fractional KdV equation, Int. J. Comput. Math., 95 (2018), 1292-1307.  doi: 10.1080/00207160.2017.1410544.  Google Scholar

[4]

Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp., 77 (2008), 699-730.  doi: 10.1090/S0025-5718-07-02045-5.  Google Scholar

[5]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.  Google Scholar

[6]

B. Cockburn and K. Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems, Numer. Math., 130 (2015), 293-314.  doi: 10.1007/s00211-014-0661-x.  Google Scholar

[7]

P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, volume 16 of Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2000.  Google Scholar

[8]

M. Fung, Kdv equation as an euler-poincare equation, Chinese J. Phys., 35 (1997), 789-796.   Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[10]

R. Hilfer, editor., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[11]

C. Huang, N. An and X. Yu, A fully discrete direct discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Anal., 97 (2018), 659-675.  doi: 10.1080/00036811.2017.1281407.  Google Scholar

[12]

C. Huang, M. Stynes and N. An, Optimal ${L}^\infty ({L}^2)$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem, BIT. Numer. Math, 58 (2018), 661-690.  doi: 10.1007/s10543-018-0707-z.  Google Scholar

[13]

C. Huang, X. Yu, C. Wang, Z. Li and N. An, A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation, Appl. Math. Comput., 264 (2015), 483-492.  doi: 10.1016/j.amc.2015.04.093.  Google Scholar

[14]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[15]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.  Google Scholar

[16]

W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617.  Google Scholar

[17]

S. Momani and A. Yıldı rım, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method, Int. J. Comput. Math., 87 (2010), 1057-1065.  doi: 10.1080/00207160903023581.  Google Scholar

[18]

D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138-1145.  doi: 10.1016/j.camwa.2008.02.015.  Google Scholar

[19]

K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995.  doi: 10.1090/S0025-5718-09-02234-0.  Google Scholar

[20]

K. Mustapha and W. McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal., 32 (2012), 906-925.  doi: 10.1093/imanum/drr027.  Google Scholar

[21]

K. Mustapha, M. Nour and B. Cockburn, Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems, Adv. Comput. Math., 42 (2016), 377-393.  doi: 10.1007/s10444-015-9428-x.  Google Scholar

[22]

I. Podlubny, Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.  Google Scholar

[23]

I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386. Dedicated to the 60th anniversary of Prof. Francesco Mainardi.  Google Scholar

[24]

J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray. Google Scholar

[25]

M. Stynes, E. O'Riordan and J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079.  doi: 10.1137/16M1082329.  Google Scholar

[26]

I. Turner, F. Liu, V. Anh and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089.  Google Scholar

[27]

L. Wei, Y. He, A. Yildirim and S. Kumar, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, ZAMM Z. Angew. Math. Mech., 93 (2013), 14-28.  doi: 10.1002/zamm.201200003.  Google Scholar

[28]

G. H. Weiss, R. Klages, G. Radons and I. M. Sokolov (eds.), Anomalous transport: Foundations and applications [book review of WILEY-VCH Verlag GmbH & Co., Weinheim, 2008], J. Stat. Phys., 135 (2009), 389-391. doi: 10.1007/s10955-009-9713-5.  Google Scholar

[29]

G. B. Witham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[30]

N. Zabusky and M. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

[31]

Q. Zhang, J. Zhang, S. Jiang and Z. Zhang, Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. Comput., 87 (2018), 693-719.  doi: 10.1090/mcom/3229.  Google Scholar

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