# American Institute of Mathematical Sciences

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

 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  January 2020 Early access  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.

Citation: Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete and 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. [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. [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. [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. [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [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. [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. [8] M. Fung, Kdv equation as an euler-poincare equation, Chinese J. Phys., 35 (1997), 789-796. [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. [10] R. Hilfer, editor., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. [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. [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. [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. [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. [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. [16] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617. [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. [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. [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. [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. [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. [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. [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. [24] J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray. [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. [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. [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. [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. [29] G. B. Witham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. [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. [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.

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. [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. [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. [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. [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [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. [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. [8] M. Fung, Kdv equation as an euler-poincare equation, Chinese J. Phys., 35 (1997), 789-796. [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. [10] R. Hilfer, editor., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. [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. [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. [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. [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. [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. [16] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617. [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. [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. [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. [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. [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. [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. [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. [24] J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray. [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. [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. [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. [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. [29] G. B. Witham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. [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. [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.
The numerical solution for Example 4.1 with $\alpha = 0.4$
$L^\infty(L^2)$ 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
 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
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
 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
$L^\infty(L^2)$ 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
 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
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
 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
$L^2$ 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
 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
$L^2$ 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
 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
 [1] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319 [2] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340 [3] Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104 [4] Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 [5] Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure and Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719 [6] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [7] Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 [8] Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 [9] Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489 [10] Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677 [11] Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811 [12] Hui Peng, Qilong Zhai. Weak Galerkin method for the Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1853-1875. doi: 10.3934/dcdsb.2021112 [13] Xiu Ye, Shangyou Zhang. A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4131-4145. doi: 10.3934/dcdsb.2020277 [14] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [15] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120 [16] Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473 [17] El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401 [18] Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 [19] S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure and Applied Analysis, 2003, 2 (3) : 277-296. doi: 10.3934/cpaa.2003.2.277 [20] Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

2020 Impact Factor: 1.327

## Metrics

• HTML views (299)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar