The weak Galerkin finite element method with or without stabilizer for the damped sine-Gordon equation and its reduced-order simulation

Senwen Deng; Minfu Feng; Xi Li; Li Zhang; Senwen Deng; Minfu Feng; Xi Li; Li Zhang

doi:10.3934/era.2025242

Electronic Research Archive

2025, Volume 33, Issue 9: 5401-5425. doi: 10.3934/era.2025242

Previous Article Next Article

Research article Special Issues

The weak Galerkin finite element method with or without stabilizer for the damped sine-Gordon equation and its reduced-order simulation

1.
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China
2.
School of Mathematics, Sichuan University, Chengdu 610066, China
3.
School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China

Received: 21 July 2025 Revised: 23 August 2025 Accepted: 29 August 2025 Published: 12 September 2025

In this work, an efficient scheme is proposed for the 2D damped sine-Gordon equation. The new scheme utilizes the unified weak Galerkin finite element method with or without stabilizer (UWG) in the spatial direction and the second-order explicit differentiation formula in the temporal direction. Both semi discrete schemes and fully discrete schemes are analyzed. First, we demonstrate the stability of the schemes. Subsequently, optimal error estimates are derived in the energy norm and the $ L^2 $ norm. Furthermore, by combining the proper orthogonal decomposition (POD) technique with the WG method, we develop a reduced-order algorithm, significantly improving computational efficiency. Finally, the theoretical results are validated by numerical experiments.
- damped sine-Gordon equation,
- weak Galerkin finite element method,
- with or without stabilizer,
- POD
Citation: Senwen Deng, Minfu Feng, Xi Li, Li Zhang. The weak Galerkin finite element method with or without stabilizer for the damped sine-Gordon equation and its reduced-order simulation[J]. Electronic Research Archive, 2025, 33(9): 5401-5425. doi: 10.3934/era.2025242

Related Papers:

Abstract

In this work, an efficient scheme is proposed for the 2D damped sine-Gordon equation. The new scheme utilizes the unified weak Galerkin finite element method with or without stabilizer (UWG) in the spatial direction and the second-order explicit differentiation formula in the temporal direction. Both semi discrete schemes and fully discrete schemes are analyzed. First, we demonstrate the stability of the schemes. Subsequently, optimal error estimates are derived in the energy norm and the $ L^2 $ norm. Furthermore, by combining the proper orthogonal decomposition (POD) technique with the WG method, we develop a reduced-order algorithm, significantly improving computational efficiency. Finally, the theoretical results are validated by numerical experiments.

References

[1]	C. L. Terng, A higher dimension generalization of the sine-Gordon equation and its soliton theory, Ann. Math., 111 (1980), 491–510. https://doi.org/10.2307/1971106 doi: 10.2307/1971106
[2]	S. Watanabe, H. S. van der Zant, S. H. Strogatz, T. P. Orlando, Dynamics of circular arrays of Josephson junctions and the discrete sine-Gordon equation, Physica D, 97 (1996), 429–470. https://doi.org/10.1016/0167-2789(96)00083-8 doi: 10.1016/0167-2789(96)00083-8
[3]	O. Nikan, Z. Avazzadeh, M. Rasoulizadeh, Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory, Nonlinear Dyn., 106 (2021), 783–813. https://doi.org/10.1007/s11071-021-06822-4 doi: 10.1007/s11071-021-06822-4
[4]	P. Grinevich, S. Novikov, Topological charge of the real periodic finite-gap sine-Gordon solutions, Commun. Pure Appl. Math., 56 (2003), 956–978. https://doi.org/10.1002/cpa.10081 doi: 10.1002/cpa.10081
[5]	J. D. Gibbon, I. N. James, I. M. Moroz, The sine-Gordon equation as a model for a rapidly rotating baroclinic fluid, Phys. Scr., 20 (1979), 402. https://doi.org/10.1088/0031-8949/20/3-4/015 doi: 10.1088/0031-8949/20/3-4/015
[6]	M. H. Uddin, U. Zaman, M. A. Arefin, M. A. Akbar, Nonlinear dispersive wave propagation pattern in optical fiber system, Chaos, Solitons Fractals, 164 (2022), 112596. https://doi.org/10.1016/j.chaos.2022.112596 doi: 10.1016/j.chaos.2022.112596
[7]	S. Duan, Y. Ma, W. Zhang, Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields, Commun. Anal. Mech., 15 (2023), 111–131. https://doi.org/10.3934/cam.2023007 doi: 10.3934/cam.2023007
[8]	F. Mirzaee, S. Rezaei, N. Samadyar, Numerical solution of two-dimensional stochastic time-fractional sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods, Eng. Anal. Boundary Elem., 127 (2021), 53–63. https://doi.org/10.1016/j.enganabound.2021.03.009 doi: 10.1016/j.enganabound.2021.03.009
[9]	Y. Yang, C. Wen, Y. Liu, H. Li, J. Wang, Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model, Commun. Anal. Mech., 16 (2024), 24–52. https://doi.org/10.3934/cam.2024002 doi: 10.3934/cam.2024002
[10]	R. Jiwari, Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine-Gordon equation, Numer. Methods Partial Differ. Equations, 37 (2021), 1965–1992. https://doi.org/10.1002/num.22636 doi: 10.1002/num.22636
[11]	M. Ratas, A. Salupere, J. Majak, Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids, Math. Model. Anal., 26 (2021), 147–169. https://doi.org/10.3846/mma.2021.12920 doi: 10.3846/mma.2021.12920
[12]	H. Zhang, X. Qian, J. Xia, S. Song, Unconditionally maximum-principle-preserving parametric integrating factor two-step Runge-Kutta schemes for parabolic sine-Gordon equations, CSIAM Trans. Appl. Math., 4 (2023), 177–224. https://doi.org/10.4208/csiam-am.SO-2022-0019 doi: 10.4208/csiam-am.SO-2022-0019
[13]	J. Y. Wang, Q. A. Huang, A family of effective structure-preserving schemes with second-order accuracy for the undamped sine-Gordon equation, Comput. Math. Appl., 90 (2021), 38–45. https://doi.org/10.1016/j.camwa.2021.03.009 doi: 10.1016/j.camwa.2021.03.009
[14]	A. Al-Taweel, J. Alkhalissi, X. Wang, The Crank-Nicolson weak Galerkin finite element methods for the sine-Gordon equation, Appl. Numer. Math., 212 (2025), 77–91. https://doi.org/10.1016/j.apnum.2025.01.016 doi: 10.1016/j.apnum.2025.01.016
[15]	J. Wang, X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103–115. https://doi.org/10.1016/j.cam.2012.10.003 doi: 10.1016/j.cam.2012.10.003
[16]	J. Wang, X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101–2126. https://doi.org/10.1090/S0025-5718-2014-02852-4 doi: 10.1090/S0025-5718-2014-02852-4
[17]	L. Zhang, M. Feng, J. Zhang, A globally divergence-free weak Galerkin method for Brinkman equations, Appl. Numer. Math., 137 (2019), 213–229. https://doi.org/10.1016/j.apnum.2018.11.002 doi: 10.1016/j.apnum.2018.11.002
[18]	X. Li, A weak Galerkin meshless method for incompressible Navier-Stokes equations, J. Comput. Appl. Math., 445 (2024), 115823. https://doi.org/10.1016/j.cam.2024.115823 doi: 10.1016/j.cam.2024.115823
[19]	D. Li, C. Wang, J. Wang, Generalized weak Galerkin finite element methods for biharmonic equations, J. Comput. Appl. Math., 434 (2023), 115353. https://doi.org/10.1016/j.cam.2023.115353 doi: 10.1016/j.cam.2023.115353
[20]	F. Huo, R. Wang, Y. Wang, R. Zhang, A locking-free weak Galerkin finite element method for linear elasticity problems, Comput. Math. Appl., 160 (2024), 181–190. https://doi.org/10.1016/j.camwa.2024.02.032 doi: 10.1016/j.camwa.2024.02.032
[21]	W. Li, F. Gao, J. Cui, A weak Galerkin finite element method for nonlinear convection-diffusion equation, Appl. Math. Comput., 461 (2024), 128315. https://doi.org/10.1016/j.amc.2023.128315 doi: 10.1016/j.amc.2023.128315
[22]	J. Zhang, X. Liu, Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D, Appl. Math. Comput., 432 (2022), 127346. https://doi.org/10.1016/j.amc.2022.127346 doi: 10.1016/j.amc.2022.127346
[23]	P. Jana, N. Kumar, B. Deka, Weak Galerkin finite element methods for semilinear Klein-Gordon equation on polygonal meshes, Comput. Appl. Math., 43 (2024), 218. https://doi.org/10.1007/s40314-024-02745-z doi: 10.1007/s40314-024-02745-z
[24]	X. Ye, S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math., 371 (2020), 112699. https://doi.org/10.1016/j.cam.2019.112699 doi: 10.1016/j.cam.2019.112699
[25]	X. Ye, S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part Ⅱ, J. Comput. Appl. Math., 394 (2021), 113525. https://doi.org/10.1016/j.cam.2021.113525 doi: 10.1016/j.cam.2021.113525
[26]	X. Ye, S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part Ⅲ, J. Comput. Appl. Math., 394 (2021), 113538. https://doi.org/10.1016/j.cam.2021.113538 doi: 10.1016/j.cam.2021.113538
[27]	Y. Feng, Y. Liu, R. Wang, S. Zhang, A stabilizer-free weak Galerkin finite element method for the Stokes equations, Adv. Appl. Math. Mech., 14 (2022), 181–201. https://doi.org/10.4208/aamm.OA-2020-0325 doi: 10.4208/aamm.OA-2020-0325
[28]	S. Wang, J. Ma, N. Du, A stabilizer-free weak Galerkin finite element method for an optimal control problem of a time fractional diffusion equation, Math. Comput. Simul., 231 (2025), 99–118. https://doi.org/10.1016/j.matcom.2024.11.019 doi: 10.1016/j.matcom.2024.11.019
[29]	X. Wang, X. Ye, S. Zhang, Weak Galerkin finite element methods with or without stabilizers, Numer. Algorithms, 88 (2021), 1361–1381. https://doi.org/10.1007/s11075-021-01079-9 doi: 10.1007/s11075-021-01079-9
[30]	K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117–148. https://doi.org/10.1007/s002110100282 doi: 10.1007/s002110100282
[31]	K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), 492–515. https://doi.org/10.1137/S0036142900382612 doi: 10.1137/S0036142900382612
[32]	J. R. Singler, New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs, SIAM. J. Numer. Anal., 52 (2014), 852–876. https://doi.org/10.1137/120886947 doi: 10.1137/120886947
[33]	J. Shen, J. R. Singler, Y. Zhang, HDG-POD reduced order model of the heat equation, J. Comput. Appl. Math., 362 (2019), 663–679. https://doi.org/10.1016/j.cam.2018.09.031 doi: 10.1016/j.cam.2018.09.031
[34]	K. Li, T. Huang, L. Li, S. Lanteri, L. Xu, B. Li, A reduced-order discontinuous Galerkin method based on POD for electromagnetic simulation, IEEE Trans. Antennas Propag., 66 (2017), 242–254. https://doi.org/10.1109/TAP.2017.2768562 doi: 10.1109/TAP.2017.2768562
[35]	S. He, H. Li, Y. Liu, A POD based extrapolation DG time stepping space-time FE method for parabolic problems, J. Math. Anal. Appl., 539 (2024), 128501. https://doi.org/10.1016/j.jmaa.2024.128501 doi: 10.1016/j.jmaa.2024.128501
[36]	L. Wang, M. Feng, The simplified weak Galerkin method with $\theta$ scheme and its reduced-order model for the elastodynamic problem on polygonal mesh, Comput. Math. Appl., 178 (2025), 19–46. https://doi.org/10.1016/j.camwa.2024.11.023 doi: 10.1016/j.camwa.2024.11.023
[37]	W. Han, L. He, F. Wang, Optimal order error estimates for discontinuous Galerkin methods for the wave equation, J. Sci. Comput., 78 (2019), 121–144. https://doi.org/10.1007/s10915-018-0755-1 doi: 10.1007/s10915-018-0755-1
[38]	H. Wang, D. Xu, J. Zhou, J. Guo, Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation, Numer. Methods Partial Differ. Equations, 37 (2021), 732–749. https://doi.org/10.1002/num.22549 doi: 10.1002/num.22549
[39]	L. Wang, Y. Tan, M. Feng, The two-grid hybrid high-order method for the nonlinear strongly damped wave equation on polygonal mesh and its reduced-order model, Appl. Numer. Math., 210 (2025), 1–24. https://doi.org/10.1016/j.apnum.2024.12.006 doi: 10.1016/j.apnum.2024.12.006
[40]	Q. Zhai, R. Wang, L. Mu, Implementation of weak Galerkin finite element methods, 2016. Available from: https://github.com/MollyRaver/WGFEMPoisson.
[41]	M. Dehghan, D. Mirzaei, The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. Methods Appl. Mech. Eng., 197 (2008), 476–486. https://doi.org/10.1016/j.cma.2007.08.016 doi: 10.1016/j.cma.2007.08.016
[42]	A. Bratsos, The solution of the two-dimensional sine-Gordon equation using the method of lines, J. Comput. Appl. Math., 206 (2007), 251–277. https://doi.org/10.1016/j.cam.2006.07.002 doi: 10.1016/j.cam.2006.07.002

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)