Skip to main content
SpringerLink
Log in

Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This paper aims at developing a dissipation-preserving, linearized, and time-stepping-varying spectral method for strongly damped nonlinear wave system in multidimensional unbounded domains \({\mathbb {R}}^d\) (d=1, 2, and 3), where the nonlocal nature is described by the mixed fractional Laplacians. Because the underlying solutions of the problem involving mixed fractional Laplacians decay slowly with certain power law at infinity, we employ the rational spectral-Galerkin method using rational basis (or mapped Gegenbauer functions) for the spatial approximation. To capture the intrinsic dissipative properties of the model equations, we combine the Crank-Nicolson scheme with exponential scalar auxiliary variable approach for the temporal discretization. Based on the rate of nonlocal energy dissipation, we design a novel time-stepping-varying strategy to enhance the efficiency of the scheme. We present the detailed implementation of the scheme, where the main building block of the stiffness matrices is based on the Laguerre-Gauss quadrature rule for the modified Bessel functions of the second kind. The existence, uniqueness, and nonlocal energy dissipation law of the fully discrete scheme are rigourously established. Numerical examples in 3D case are carried out to demonstrate the accuracy and efficiency of the scheme. Finally, we simulate the nonlinear behaviors of 2D/3D dissipative vector solitary waves for damped sine-Gordon system I, for damped sine-Gordon system II, and for damped Klein–Gordon system to provide a deeper understanding of nonlocal physics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data Availability

This manuscript has no associated data.

References

  1. Abdumalikov, A.A., Alfimov, G.L., Malishevskii, A.S.: Nonlocal electrodynamics of Josephson vortices in superconducting circuits. Supercond. Sci. Technol. 22, 023001 (2009)

    Article  Google Scholar 

  2. Akhatov, ISh., Baikov, V.A., Khusnutdinova, K.R.: Non-linear dynamics of coupled chains of particles. J. Appl. Math. Mech. 59, 353–361 (1995)

    Article  MathSciNet  Google Scholar 

  3. Alfimov, G.L., Medvedeva, E.V.: Moving nonradiating kinks in nonlocal \(\varphi ^4\) and \(\varphi ^4-\varphi ^6\) models. Phys. Rev. E 84, 056606 (2011)

    Article  Google Scholar 

  4. Belenchia, A., Benincasa, D., Liberati, S., Marin, F., Marino, F., Ortolan, A.: Tests of quantum-gravity-induced nonlocality via optomechanical experiments. Phys. Rev. D 95, 026012 (2017)

    Article  MathSciNet  Google Scholar 

  5. Boyanovsky, D., Destri, C., de Vega, H.J.: Approach to thermalization in the classical \(\varphi ^4\) theory in 1+1 dimensions: Energy cascades and universal scaling. Phys. Rev. D 69, 045003 (2004)

    Article  Google Scholar 

  6. Braun, O.M., Kivshar, Y.S.: Nonlinear dynamics of the Frenkel–Kontorova model. Phys. Rep. 306, 1–108 (1998)

    Article  MathSciNet  Google Scholar 

  7. Cai, W., Zhang, H., Wang, Y.: Dissipation-preserving spectral element method for damped seismic wave equations. J. Comput. Phys. 350, 260–279 (2017)

    Article  MathSciNet  Google Scholar 

  8. Calcagni, G., Modesto, L.: Nonlocal quantum gravity and M-theory. Phys. Rev. D 91, 124059 (2015)

    Article  MathSciNet  Google Scholar 

  9. Carreño, A., Vidal-Ferrandiz, A., Ginestar, D., Verdu, G.: Adaptive time-step control for modal methods to integrate the neutron diffusion equation. Nucl. Eng. Technol. 53, 399–413 (2021)

    Article  Google Scholar 

  10. Chen, S., Shen, J., Wang, L.: Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. J. Sci. Comput. 74, 1286–1313 (2018)

    Article  MathSciNet  Google Scholar 

  11. Chen, W., Wang, X., Yan, Y., Zhang, Z.: A second order BDF numerical scheme with variable steps for the Cahn–Hilliard equation. SIAM J. Numer. Anal. 57, 495–525 (2019)

    Article  MathSciNet  Google Scholar 

  12. Conway, J.B.: Functions of One Complex Variable I. Springer, Berlin (1994)

    Google Scholar 

  13. D’Elia, M., Du, Q., Glusa, C., Gunzburger, M., Tian, X., Zhou, Z.: Numerical methods for nonlocal and fractional models. Acta Numerica 29, 1–124 (2020)

    Article  MathSciNet  Google Scholar 

  14. Du, Q.: Nonlocal Modeling, Analysis, and Computation, SIAM, 2019

  15. Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series, and Products, 8th edn. Academic Press, New York (2014)

    MATH  Google Scholar 

  16. Grønbech-Jensen, N., Blackburn, J.A., Samuelsen, M.R.: Phase locking between Fiske and flux-flow modes in coupled sine-Gordon systems. Phys. Rev. B 53, 12364 (1996)

    Article  Google Scholar 

  17. Guo, S., Mei, L., Li, C., Yan, W., Gao, J.: IMEX Hermite-Galerkin spectral schemes with adaptive time stepping for the coupled nonlocal Gordon-type systems in multiple dimensions. SIAM J. Sci. Comput. 43, B1133–B1163 (2021)

    Article  MathSciNet  Google Scholar 

  18. Han, W., Gao, J., Zhang, Y., Xu, W.: Well-posedness of the diffusive-viscous wave equation arising in geophysics. J. Math. Anal. Appl. 486, 123914 (2020)

    Article  MathSciNet  Google Scholar 

  19. Hayashi, N., Naumkin, P.I.: A system of quadratic nonlinear Klein–Gordon equations in 2d. J. Differ. Equ. 254, 3615–3646 (2013)

    Article  MathSciNet  Google Scholar 

  20. Hu, D., Cai, W., Song, Y., Wang, Y.: A fourth-order dissipation-preserving algorithm with fast implementation for space fractional nonlinear damped wave equations. Commun. Nonlinear. Sci. Numer. Simul. 91, 105432 (2020)

    Article  MathSciNet  Google Scholar 

  21. Jiang, C., Wang, Y., Cai, W.: A linearly implicit energy-preserving exponential integrator for the nonlinear Klein–Gordon equation. J. Comput. Phys. 419, 109690 (2020)

    Article  MathSciNet  Google Scholar 

  22. Khusnutdinova, K.R., Pelinovsky, D.E.: On the exchange of energy in coupled Klein-Gordon equations. Wave Motion 38, 1–10 (2003)

    Article  MathSciNet  Google Scholar 

  23. Khosravian-Arab, H., Dehghan, M., Eslahchi, M.R.: Fractional spectral and pseudo-spectral methods in unbounded domains: theory and applications. J. Comput. Phys. 338, 527–566 (2017)

    Article  MathSciNet  Google Scholar 

  24. Khosravian-Arab, H., Dehghan, M., Eslahchi, M.R.: Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications. J. Comput. Phys. 299, 526–560 (2015)

    Article  MathSciNet  Google Scholar 

  25. Li, X., Shen, J., Rui, H.: Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math. Comput. 88, 2047–2068 (2019)

    Article  MathSciNet  Google Scholar 

  26. Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M.M., Ainsworth, M., Karniadakis, G.E.: What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009 (2020)

    Article  MathSciNet  Google Scholar 

  27. Lischke, A., Zayernouri, M., Karniadakis, G.: A Petrov–Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. SIAM J. Sci. Comput. 39, A922–A946 (2017)

    Article  MathSciNet  Google Scholar 

  28. Liu, Z., Li, X.: The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing. SIAM J. Sci. Comput. 42, B630–B655 (2020)

    Article  MathSciNet  Google Scholar 

  29. Mao, Z., Shen, J.: Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput. 39, A1928–A1950 (2017)

    Article  MathSciNet  Google Scholar 

  30. Pnevmatikos, S.: Soliton dynamics of hydrogen-bonded networks: a mechanism for proton conductivity. Phys. Rev. Lett. 60, 1534–1537 (1988)

    Article  Google Scholar 

  31. Shen, J.: Efficient spectral-Galerkin method I. Direct solvers for second-and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)

    Article  MathSciNet  Google Scholar 

  32. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474–506 (2019)

    Article  MathSciNet  Google Scholar 

  33. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  Google Scholar 

  34. Sheng, C., Shen, J., Tang, T., Wang, L.L., Yuan, H.: Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains. SIAM J. Numer. Anal. 58, 2435–2464 (2020)

    Article  MathSciNet  Google Scholar 

  35. Szabo, T.L.: Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96, 491–500 (1994)

    Article  Google Scholar 

  36. Tang, T., Wang, L.L., Yuan, H., Zhou, T.: Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains. SIAM J. Sci. Comput. 42, A585–A611 (2020)

    Article  MathSciNet  Google Scholar 

  37. Tang, T., Yuan, H., Zhou, T.: Hermite spectral collocation methods for fractional PDEs in unbounded domains. Commun. Comput. Phys. 24, 1143–1168 (2018)

    Article  MathSciNet  Google Scholar 

  38. Yomosa, S.: Soliton excitations in deoxyribonucleic acid (DNA) double helices. Phys. Rev. A 27, 2120–2125 (1983)

    Article  MathSciNet  Google Scholar 

  39. Zhang, H., Jiang, X., Zeng, F., Karniadakis, G.E.: A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction–diffusion equations. J. Comput. Phys. 405, 109141 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the referees for the constructive comments which led to an improved version.

Funding

The project is supported by National Key R &D Program of China (2020YFA0713400 and 2020YFA0713401), NSF of China (12271427, 11901489, and 12171385), and NSF of ShaanXi Province (2020JQ-008).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Shimin Guo, Wenjing Yan & Liquan Mei

  2. School of Sciences, Xi’an University of Technology, Xi’an, 710054, China

    Can Li

Authors
  1. Shimin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenjing Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Can Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Liquan Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. The manuscript was written by Shimin Guo and Wenjing Yan and all authors commented on the present version of the manuscript.

Corresponding author

Correspondence to Wenjing Yan.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, S., Yan, W., Li, C. et al. Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains. J Sci Comput 93, 53 (2022). https://doi.org/10.1007/s10915-022-02008-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-02008-1

Keywords

Search

Navigation