A linearly implicit local energy-dissipative method for the generalized nonlinear wave equation

Yulian Yi; Yuchen Yin; Mingfa Fei; Yulian Yi; Yuchen Yin; Mingfa Fei

doi:10.3934/nhm.2026015

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 324-348. doi: 10.3934/nhm.2026015

Previous Article Next Article

Research article

A linearly implicit local energy-dissipative method for the generalized nonlinear wave equation

1.
School of Mathematics, Changsha University, Changsha 410022, China
2.
Teachers College, Columbia University, New York 10027, USA

Received: 05 December 2025 Revised: 03 March 2026 Accepted: 04 March 2026 Published: 18 March 2026

In this paper, we propose a linearly implicit scheme that preserves the local energy dissipation property for the generalized nonlinear wave equation. By employing the energy quadratization approach, we reformulated the original equation into an equivalent form. A semi-discrete structure-preserving system was constructed via finite difference discretization in space. Subsequently, we derived a fully discretized scheme using the Crank-Nicolson method combined with the extrapolation technique, ensuring the preservation of local energy dissipation law. Under appropriate boundary conditions, such as homogeneous Dirichlet or periodic boundary conditions, the proposed method also maintained the global energy dissipation law. Furthermore, the unique solvability, fast implementation, and convergence theorem of the discrete scheme were analyzed rigorously. Numerical experiments are presented to validate our theoretical results, demonstrating that the proposed scheme outperforms traditional methods in terms of stability and efficiency.
- nonlinear wave equation,
- linearly implicit scheme,
- structure-preserving method,
- local energy dissipation law,
- convergence analysis
Citation: Yulian Yi, Yuchen Yin, Mingfa Fei. A linearly implicit local energy-dissipative method for the generalized nonlinear wave equation[J]. Networks and Heterogeneous Media, 2026, 21(1): 324-348. doi: 10.3934/nhm.2026015

Related Papers:

Abstract

In this paper, we propose a linearly implicit scheme that preserves the local energy dissipation property for the generalized nonlinear wave equation. By employing the energy quadratization approach, we reformulated the original equation into an equivalent form. A semi-discrete structure-preserving system was constructed via finite difference discretization in space. Subsequently, we derived a fully discretized scheme using the Crank-Nicolson method combined with the extrapolation technique, ensuring the preservation of local energy dissipation law. Under appropriate boundary conditions, such as homogeneous Dirichlet or periodic boundary conditions, the proposed method also maintained the global energy dissipation law. Furthermore, the unique solvability, fast implementation, and convergence theorem of the discrete scheme were analyzed rigorously. Numerical experiments are presented to validate our theoretical results, demonstrating that the proposed scheme outperforms traditional methods in terms of stability and efficiency.

References

[1]	D. Hu, Fully decoupled and high-order linearly implicit energy-preserving RK-GSAV methods for the coupled nonlinear wave equation, J. Comput. Appl. Math., 445 (2024), 115836. https://doi.org/10.1016/j.cam.2024.115836 doi: 10.1016/j.cam.2024.115836
[2]	L. Liu, C. Li, Coupled sine-Gordon systems in DNA dynamics, Adv. Math. Phys., 2018 (2018), 4676281. https://doi.org/10.1155/2018/4676281 doi: 10.1155/2018/4676281
[3]	S. Joseph, New traveling wave exact solutions to the coupled Klein-Gordon system of equations, Partial Differ. Equations Appl. Math., 5 (2022), 100208. https://doi.org/10.1016/j.padiff.2021.100208 doi: 10.1016/j.padiff.2021.100208
[4]	B. Josephson, Supercurrents through barriers, Adv. Phys., 14 (2006), 419–451. https://doi.org/10.1080/00018736500101091 doi: 10.1080/00018736500101091
[5]	Q. Sheng, A. Khaliq, D. Voss, Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Simul., 68 (2005), 355–373. https://doi.org/10.1016/j.matcom.2005.02.017 doi: 10.1016/j.matcom.2005.02.017
[6]	S. Dong, Z. Wyatt, Stability of a coupled wave-Klein-Gordon system with quadratic nonlinearities, J. Differ. Equations, 269 (2020), 7470–7497. https://doi.org/10.1016/j.jde.2020.05.019 doi: 10.1016/j.jde.2020.05.019
[7]	M. Grote, A. Schneebeli, D. Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal., 44 (2006), 2408–2431. https://doi.org/10.1137/05063194X doi: 10.1137/05063194X
[8]	G. Richter, An explicit finite element method for the wave equation, Appl. Numer. Math., 16 (1994), 65–80. https://doi.org/10.1016/0168-9274(94)00048-4 doi: 10.1016/0168-9274(94)00048-4
[9]	A. Pleshkevich, D. Vishnevskiy, V. Lisitsa, Sixth-order accurate pseudo-spectral method for solving one-way wave equation, Appl. Math. Comput., 359 (2019), 34–51. https://doi.org/10.1016/j.amc.2019.04.029 doi: 10.1016/j.amc.2019.04.029
[10]	K. Djidjeli, W. Price, E. Twizell, Q. Cao, A linearized implicit pseudo-spectral method for some model equations: The regularized long wave equations, Commun. Numer. Methods Eng., 19 (2003), 847–863. https://doi.org/10.1002/cnm.635 doi: 10.1002/cnm.635
[11]	Y. Zhao, X. Gu, A low-rank algorithm for strongly damped wave equations with visco-elastic damping and mass terms, ESAIM: Math. Model. Numer. Anal., 59 (2025), 1747–1761. https://doi.org/10.1051/m2an/2025042 doi: 10.1051/m2an/2025042
[12]	W. Luo, X. Gu, B. Carpentieri, J. Guo, A Bernoulli-barycentric rational matrix collocation method with preconditioning for a class of evolutionary PDEs, Numer. Linear Algebra Appl., 32 (2025), e70007. https://doi.org/10.1002/nla.70007 doi: 10.1002/nla.70007
[13]	Y. Chen, H. Zhu, S. Song, Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 181 (2011), 1231–1241. https://doi.org/10.1016/j.cpc.2010.03.009 doi: 10.1016/j.cpc.2010.03.009
[14]	C. Jiang, J. Sun, H. Li, Y. Wang, A fourth-order AVF method for the numerical integration of sine-Gordon equation, Appl. Math. Comput., 313 (2017), 144–158. https://doi.org/10.1016/j.amc.2017.05.055 doi: 10.1016/j.amc.2017.05.055
[15]	B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics, Cambridge university press, Cambridge, 2004. https://doi.org/10.1016/j.amc.2017.05.055
[16]	Y. Li, X. Wu, Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems, SIAM J. Sci. Comput., 38 (2016), A1876–A1895. https://doi.org/10.1137/15M1023257 doi: 10.1137/15M1023257
[17]	L. Brugnano, C. Zhang, D. Li, A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 33–49. https://doi.org/10.1016/j.cnsns.2017.12.018 doi: 10.1016/j.cnsns.2017.12.018
[18]	K. Feng, M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer, Berlin, 2010. Available from: https://link.springer.com/book/10.1007/978-3-642-01777-3.
[19]	W. Bao, X. Zhao, A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein-Gordon-Schrödinger equations in the nonrelativistic limit regime, Numer. Math., 135 (2007), 833–873. https://doi.org/10.1007/s00211-016-0818-x doi: 10.1007/s00211-016-0818-x
[20]	Y. Gong, Q. Hong, Q. Wang, Supplementary variable method for thermodynamically consistent partial differential equations, Comput. Methods Appl. Mech. Eng., 381 (2021), 113746. https://doi.org/10.1016/j.cma.2021.113746 doi: 10.1016/j.cma.2021.113746
[21]	F. Yin, Z. Xu, Y. Fu, Novel high-order explicit energy-preserving schemes for NLS-type equations based on the Lie-group method, Math. Comput. Simul., 225 (2024), 570–585. https://doi.org/10.1016/j.matcom.2024.05.029 doi: 10.1016/j.matcom.2024.05.029
[22]	X. Li, L. Zhang, High-order conservative energy quadratization schemes for the Klein-Gordon-Schrödinger equation, Adv. Comput. Math., 48 (2022), 41. https://doi.org/10.1007/s10444-022-09962-2 doi: 10.1007/s10444-022-09962-2
[23]	Z. Zhang, J. Shen, Efficient structure preserving schemes for the Klein-Gordon-Schrödinger equations, J. Sci. Comput., 89 (2021), 47. https://doi.org/10.1007/s10915-021-01649-y doi: 10.1007/s10915-021-01649-y
[24]	Q. Hong, J. Li, Q. Wang. Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations, Appl. Math. Lett., 110 (2020), 106576. https://doi.org/10.1016/j.aml.2020.106576 doi: 10.1016/j.aml.2020.106576
[25]	E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Science, London, 2006. Available from: https://link.springer.com/book/10.1007/3-540-30666-8.
[26]	H. Li, Y. Wang, M. Qin, A sixth order averaged vector field method, J. Comput. Math., 34 (2016), 479–498. https://doi.org/10.4208/jcm.1601-m2015-0265 doi: 10.4208/jcm.1601-m2015-0265
[27]	Q. Hong, Y. Wang, J. Wang, Optimal error estimate of a linear Fourier pseudo-spectral scheme for two-dimensional Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 468 (2018), 817–838. https://doi.org/10.1016/j.jmaa.2018.08.045 doi: 10.1016/j.jmaa.2018.08.045
[28]	J. Wang, A. Xiao, Conservative Fourier spectral method and numerical investigation of space fractional Klein-Gordon-Schrödinger equations, Appl. Math. Comput., 350 (2019), 348–365. https://doi.org/10.1016/j.amc.2018.12.046 doi: 10.1016/j.amc.2018.12.046
[29]	J. Macías-Díaz, A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, J. Comput. Phys., 351 (2017), 40–58. https://doi.org/10.1016/j.jcp.2017.09.028 doi: 10.1016/j.jcp.2017.09.028
[30]	J. Macías-Díaz, A numerically efficient dissipation-preserving implicit method for a nonlinear multidimensional fractional wave equation, J. Sci. Comput., 77 (2018), 1–26. https://doi.org/10.1007/s10915-018-0692-z doi: 10.1007/s10915-018-0692-z
[31]	J. Macías-Díaz, A. Hendy, R. Staelen, A compact fourth-order in space energy-preserving method for {R}iesz space-fractional nonlinear wave equations, Appl. Math. Comput., 325 (2018), 1–14. https://doi.org/10.1016/j.amc.2017.12.002 doi: 10.1016/j.amc.2017.12.002
[32]	D. Hu, W. Cai, Y. Song, Y. Wang, A fourth-order dissipation-preserving algorithm with fast implementation for space fractional nonlinear damped wave equations, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105432. https://doi.org/10.1016/j.cnsns.2020.105432 doi: 10.1016/j.cnsns.2020.105432
[33]	D. Hu, Y. Gong, Y. Wang, On convergence of a structure preserving difference scheme for two–dimensional space–fractional nonlinear Schrödinger equation and its fast implementation, Comput. Math. Appl., 98 (2021), 10–23. https://doi.org/10.1016/j.camwa.2021.06.018 doi: 10.1016/j.camwa.2021.06.018
[34]	Z. Sun, Y. Liu, B. Yin, H. Li, Fast structure-preserving difference algorithm for 2D nonlinear space-fractional wave models, Comput. Math. Appl., 123 (2022), 40–58. https://doi.org/10.1016/j.camwa.2022.07.020 doi: 10.1016/j.camwa.2022.07.020
[35]	N. Wang, M. Li, C. Huang, Unconditional energy dissipation and error estimates of the SAV Fourier spectral method for nonlinear fractional generalized wave equation, J. Sci. Comput., 88 (2021), 19. https://doi.org/10.1007/s10915-021-01534-8 doi: 10.1007/s10915-021-01534-8
[36]	J. Yan, Z. Zhang, New energy-preserving schemes using Hamiltonian boundary value and Fourier pseudospectral methods for the numerical solution of "good" Boussinesq equation, Comput. Phys. Commun., 201 (2016), 33–42. https://doi.org/10.1016/j.cpc.2015.12.013 doi: 10.1016/j.cpc.2015.12.013
[37]	Y. Gong, J. Zhao, X. Yang, Q. Wang, Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities, SIAM J. Sci. Comput., 40 (2018), B138–B167. https://doi.org/10.1137/17M1111759 doi: 10.1137/17M1111759
[38]	X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294–316. https://doi.org/10.1016/j.jcp.2016.09.029 doi: 10.1016/j.jcp.2016.09.029
[39]	J. Shen, J. Xu, J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407–416. https://doi.org/10.1016/j.jcp.2017.10.021 doi: 10.1016/j.jcp.2017.10.021
[40]	J. Shen, J. Xu, Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows, SIAM J. Numer. Anal., 56 (2018), 2895–2912. https://doi.org/10.1137/17M1159968 doi: 10.1137/17M1159968
[41]	Y. Fu, D. Hu, Y. Wang, High-order structure-preserving algorithms for the multi-dimensional fractional nonlinear Schrödinger equation based on the SAV approach, Math. Comput. Simul., 185 (2021), 238–255. https://doi.org/10.1016/j.matcom.2020.12.025 doi: 10.1016/j.matcom.2020.12.025
[42]	Y. Fu, Z. Xu, W. Cai, Y. Wang, An efficient energy-preserving method for the two-dimensional fractional Schrödinger equation, Appl. Numer. Math., 165 (2021), 232–247. https://doi.org/10.1016/j.apnum.2021.02.010 doi: 10.1016/j.apnum.2021.02.010
[43]	C. Jiang, Y. Wang, Y. Gong, Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation, Appl. Numer. Math., 151 (2020), 85–97. https://doi.org/10.1016/j.apnum.2019.12.016 doi: 10.1016/j.apnum.2019.12.016
[44]	Z. Liu, X. Li, The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing, SIAM J. Sci. Comput., 42 (2019), B630–B655. https://doi.org/10.1137/19M1305914 doi: 10.1137/19M1305914
[45]	Y. Wang, B. Wang, M. Qin, Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A-Math., 51 (2008), 2115–2136. https://doi.org/10.1007/s11425-008-0046-7 doi: 10.1007/s11425-008-0046-7
[46]	C. Jiang, W. Cai, Y. Wang, A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach, J. Sci. Comput., 80 (2019), 1629–1655. https://doi.org/10.1007/s10915-019-01001-5 doi: 10.1007/s10915-019-01001-5
[47]	Y. Gong, J. Cai, Y. Wang, Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), 80–102. https://doi.org/10.1016/j.jcp.2014.09.001 doi: 10.1016/j.jcp.2014.09.001
[48]	Y. Li, X. Wu, General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 301 (2015), 141–166. https://doi.org/10.1016/j.jcp.2015.08.023 doi: 10.1016/j.jcp.2015.08.023
[49]	Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Methods, International Academic Publishers, Beijing, 1990.
[50]	E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren, et al., Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method, J. Comput. Phys., 231 (2012), 6770–6789. https://doi.org/10.1016/j.jcp.2012.06.022 doi: 10.1016/j.jcp.2012.06.022
[51]	S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), 473–499. https://doi.org/10.1006/jcph.1999.6372 doi: 10.1006/jcph.1999.6372
[52]	C. Schober, T. Wlodarczyk, Dispersive properties of multisymplectic integrators, J. Comput. Phys., 227 (2008), 5090–5104. https://doi.org/10.1016/j.jcp.2008.01.026 doi: 10.1016/j.jcp.2008.01.026
[53]	W. Cai, C. Jiang, Y. Wang, Y. Song, Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions, J. Comput. Phys., 395 (2019), 166–185. https://doi.org/10.1016/j.jcp.2019.05.048 doi: 10.1016/j.jcp.2019.05.048

Reader Comments

Your name:*

Email:*
© 2026 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)