A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System

doi:10.1007/s42967-021-00178-7

Abstract

Abstract: The purpose of this paper is to develop a hybridized discontinuous Galerkin (HDG) method for solving the Ito-type coupled KdV system. In fact, we use the HDG method for discretizing the space variable and the backward Euler explicit method for the time variable. To linearize the system, the time-lagging approach is also applied. The numerical stability of the method in the sense of the L₂ norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet boundary conditions. Numerical experiments confirm that the HDG method is capable of solving the system efficiently. It is observed that the best possible rate of convergence is achieved by the HDG method. Also, it is being illustrated numerically that the corresponding conservation laws are satisfied for the approximate solutions of the Ito-type coupled KdV system. Thanks to the numerical experiments, it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.

Key words: Hybridized discontinuous Galerkin (HDG) method, Stability analysis, Ito-type coupled KdV system, Conservation laws

CLC Number:

Shima Baharlouei, Reza Mokhtari, Nabi Chegini. A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1351-1373.

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References

1. Akbari, R., Mokhtari, R.:A new compact finite difference method for solving the generalized long wave equation. Numer. Funct. Anal. Optim. 35(2), 133-152 (2014)
2. Atkinson, K.E., Han, W.:Numerical Solution of Ordinary Differential Equations. John Wiley and Sons, Inc., New Jersey (2009)
3. Başhan, A.:An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods. Comput. Appl. Math. 39(80), 1-23 (2020)
4. Bona, J.L., Chen, H., Karakashian, O., Wise, M.M.:Finite element methods for a system of dispersive equations. J. Sci. Comput. 77(3), 1371-1401 (2018)
5. Buffa, A., Ortner, C.:Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 29(4), 827-855 (2009)
6. Cao, W., Fei, J., Ma, Z., Liu, Q.:Bosonization and new interaction solutions for the coupled Korteweg-de Vries system. Waves Random Complex Media 30(1), 130-141 (2020)
7. Castillo, P., Gomez, S.:Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrödinger equations. Appl. Math. Comput. 371, 124950 (2020)
8. Castillo, P., Gomez, S.:Conservative local discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction. Numer. Algor. 84(1), 407-425 (2020)
9. Chegini, N., Stevenson, R.:An adaptive wavelet method for semi-linear first-order system least squares. J. Comput. Meth. Appl. Math. 15(4), 439-463 (2015)
10. Chen, Y., Song, S., Zhu, H.:Multi-symplectic methods for the Ito-type coupled KdV equation. Appl. Math. Comput. 218(9), 5552-5561 (2012)
11. Cockburn, B., Gopalakrishnan, J., Lazarov, R.:Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319-1365 (2009)
12. Cockburn, B., Shu, C.-W.:The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440-2463 (1998)
13. Dong, B.:Optimally convergent HDG method for third-order Korteweg-de Vries type equations. J. Sci. Comput. 73(2/3), 712-735 (2017)
14. Drinfel'd, V.G., Sokolov, V.V.:Lie algebras and equations of Korteweg-de Vries type. J. Sov. Math. 30, 1975-2036 (1975)
15. Gear, J.A., Grimshaw, R.:Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70(3), 235-258 (1984)
16. Guha-Roy, C.:Solution of coupled KdV-type equations. Int. J. Theor. Phys. 29(8), 863-866 (1990)
17. Hirota, R., Satsuma, J.:Soliton solutions of a coupled Korteweg-de Vries equation. Phys. Lett. A 85(8/9), 407-408 (1981)
18. Ito, M.:Symmetries and conservation laws of a coupled nonlinear wave equation. Phys. Lett. A 91(7), 335-338 (1982)
19. Kirby, R.M., Sherwin, S.J., Cockburn, B.:To CG or to HDG:a comparative study. J. Sci. Comput. 51(1), 183-212 (2012)
20. Leng, H.T., Chen, Y.P.:Adaptive hybridizable discontinuous Galerkin methods for nonstationary convection diffusion problems. Adv. Comput. Math. 46(50), 1-23 (2020)
21. Lou, S.Y., Tong, B., Hu, H.C., Tang, X.Y.:Coupled KdV equations derived from two-layer fluids. J. Phys. A 39(3), 513-527 (2006)
22. Luo, D.M., Huang, W.Z., Qiu, J.X.:An hybrid LDG-HWENO scheme for KdV-type equations. J. Comput. Phys. 313, 754-774 (2016)
23. Mirza, A., ul Hassan, M.:Bilinearization and soliton solutions of the supersymmetric coupled KdV equation. Theor. Math. Phys. 202(1), 11-16 (2020)
24. Mogorosi, E.T., Muatjetjeja, B., Khalique, C.M.:Conservation laws for a generalized Ito-type coupled KdV system. Bound. Value Probl. 2012(1), 1-7 (2012)
25. Mohammadi, M., Mokhtari, R., Schaback, R.:A meshless method for solving the 2D Brusselator reaction-diffusion system. Comput. Model. Eng. Sci. 101(2), 113-138 (2014)
26. Mokhtari, R., Isvand, D., Chegini, N.G., Salaripanah, A.:Numerical solution of the Schrödinger equations by using Delta-shaped basis functions. Nonlinear Dyn. 74(1/2), 77-93 (2013)
27. Mokhtari, R., Mohseni, M.:A meshless method for solving mKdV equation. Comput. Phys. Commun. 183(6), 1259-1268 (2012)
28. Qiu, L., Deng, W., Hesthaven, J.S.:Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes. J. Comput. Phys. 298, 678-694 (2015)
29. Saad, Y., Sosonkina, M.:Distributed Schur complement techniques for general sparse linear systems. SIAM J. Sci. Comput. 21(4), 1337-1356 (1999)
30. Samii, A., Panda, N., Michoski, C., Dawson, C.:A hybridized discontinuous Galerkin method for nonlinear Korteweg-de Vries equation. J. Sci. Comput. 68(1), 191-212 (2016)
31. Sánchez, M.A., Ciuca, C., Nguyen, N.C., Peraire, J., Cockburn, B.:Symplectic Hamiltonian HDG methods for wave propagation phenomena. J. Comput. Phys. 350, 951-973 (2017)
32. Wang, S., Yuan, J., Deng, W., Wu, Y.:A hybridized discontinuous Galerkin method for 2D fractional convection-diffusion equations. J. Sci. Comput. 68(2), 826-847 (2016)
33. Whitham, G.B.:Linear and Nonlinear Waves. Wiley, New York (1974)
34. Xie, S.S., Yi, S.C.:A conservative compact finite difference scheme for the coupled Schrödinger-KdV equations. Adv. Comput. Math. 46(1), 1-22 (2020)
35. Xu, Y., Shu, C.-W.:Local discontinuous Galerkin methods for three class of nonlinear wave equations. J. Comput. Math. 22, 250-274 (2004)
36. Xu, Y., Shu, C.-W.:Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Comput. Methods Appl. Mech. Eng. 195(25/26/27/28), 3430-3447 (2006)
37. Yan, J., Shu, C.-W.:A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769-791 (2002)
38. Yeganeh, S., Mokhtari, R., Hesthaven, J.S.:A local discontinuous Galerkin method for two-dimensional time fractional diffusion equations. Commun. Appl. Math. Comput. 2, 689-709 (2020)
39. Yeganeh, S., Mokhtari, R., Hesthaven, J.S.:Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method. BIT Numer. Math. 57(3), 685-707 (2017)
40. Yu, J.P., Sun, Y.L., Wang, F.D.:N-soliton solutions and long-time asymptotic analysis for a generalized complex Hirota-Satsuma coupled KdV equation. Appl. Math. Lett. 106, 106370 (2020)

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