Skip to main content
SpringerLink
Log in

The Pseudospectral Method and Discrete Spectral Analysis

  • Chapter
  • First Online:
Applied Wave Mathematics

Abstract

One of the focal points of the research at the Centre for Nonlinear Studies is related to the numerical simulation of the emergence, propagation and interaction of solitary waves and solitons in nonlinear dispersive media. Based on the discrete Fourier transform the pseudospectral method can be used for the numerical integration of the model equations, and the Fourier transform related discrete spectral characteristics for the analysis of the numerical results. The latter approach is called discrete spectral analysis. The main advantage of the pseudospectral method compared with the finite difference method is related to the computational costs. On the other hand, the Fourier transform related spectral characteristics carry additional information about the internal structure of the waves, which can be used for the analysis of the time-space behaviour of the numerical solutions. In the present paper several practical aspects of the application of the Fourier transform based pseudospectral method for the numerical integration of equations of different types are discussed and several examples of applications of the discrete spectral analysis are introduced.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bao, W., Chern, I.L., Lim, F.Y.: Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates. J. Comput. Phys. 219(2), 836–854 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bao, W., Lim, F.Y.: Computing ground states of spin-1 Bose–Einstein condensates by the normalized gradient flow. SIAM J. Sci. Comput. 30(4), 1925–1948 (2008)

    Article  MathSciNet  Google Scholar 

  3. Bao, W., Yang, L.: Efficient and accurate numerical methods for the Klein–Gordon–Schrödinger equations. J. Comput. Phys. 225(2), 1863–1893 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Billingham, J., King, A.C.: Wave Motion. Cambridge University Press (2000)

    Google Scholar 

  5. Bountis, T., Starmer, C.F., Bezerianos, A.: Stationary pulses and wave front formation in an excitable medium. Progr. Theoret. Phys. Suppl. 139, 12–33 (2000)

    Article  Google Scholar 

  6. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2000)

    Google Scholar 

  7. Boyd, J.P.: Fourier pseudospectral method with Kepler mapping for travelling waves with discontinuous slope: Application to corner waves of the Ostrovsky–Hunter equation and equatorial Kelvin waves in the four-mode approximation. Appl. Math. Comput. 177(1), 289–299 (2006)

    Article  MathSciNet  Google Scholar 

  8. Bracewell, R.N.: The Fast Fourier Transform and its Applications. McGraw-Hill, New York (1978)

    Google Scholar 

  9. Cabral, M., Rosa, R.: Chaos for a damped and forced KdV equation. Physica D 192, 265–278 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, J.B.: A multisymplectic pseudospectral method for seismic modeling. Appl. Math. Comput. 186(2), 1612–1616 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dai, H.H., Huo, Y.: Solitary shock waves and other travelling waves in a general compressible hyperelastic rod. Proc. R. Soc. Lond. A 456, 331–363 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Darvishi, M.T.: Spectral collocation solution of a generalized Hirota–Satsuma coupled KdV equation. Int. J. Comput. Math. 84(4), 541–551 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Drazin, P.G., Johnson, R.S.: Solitons: an Introduction. Cambridge University Press (1989)

    Google Scholar 

  14. Engelbrecht, J.: An Introduction to Asymmetric Solitary Waves. Longman, Harlow (1991)

    MATH  Google Scholar 

  15. Engelbrecht, J.: Nonlinear Wave Dynamics: Complexity and Simplicity. Kluwer, Dordrecht (1997)

    MATH  Google Scholar 

  16. Engelbrecht, J., Berezovski, A., Pastrone, F., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85, 4127–4141 (2005)

    Article  Google Scholar 

  17. Engelbrecht, J., Berezovski, A., Salupere, A.: Nonlinear deformation waves in solids and dispersion. Wave Motion 44(6), 493–500 (2007)

    Article  MathSciNet  Google Scholar 

  18. Engelbrecht, J., Khamidullin, Y.: On the possible amplification of nonlinear seismic waves. Phys. Earth Planet. Inter. 50, 39–45 (1988)

    Article  Google Scholar 

  19. Engelbrecht, J., Pastrone, F.: Waves in microstructured solids with strong nonlinearities in microscale. Proc. Estonian Acad. Sci. Phys. Math. 52(1), 12–20 (2003)

    MATH  MathSciNet  Google Scholar 

  20. Engelbrecht, J., Peipman, T.: Nonlinear waves in a layer with energy influx. Wave Motion 16(2), 173–181 (1992)

    Article  Google Scholar 

  21. Randrüüt, M., Salupere, A., Engelbrecht, J.: On modelling wave motion in microstructured solids. Proc. Estonian Acad. Sci. 58(4), (2009)

    Google Scholar 

  22. Engelbrecht, J., Salupere, A.: On the problem of periodicity and hidden solitons for the KdV model. Chaos 15, 015114 (2005)

    Article  Google Scholar 

  23. Engelbrecht, J., Salupere, A., Kalda, J., Maugin, G.A.: Discrete spectral analysis for solitary waves. In: Guran, A., Maugin, G.A., Engelbrecht, J., Werby, M. (eds.) Acoustic Interactions with Submerged Elastic Structures. Part II: Propagation, Ocean Acoustics and Scattering, Series on Stability, Vibration and Control of Systems, Series B, Vol. 5, pp. 1–40. World Scientific, Singapore (2001)

    Chapter  Google Scholar 

  24. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press (1998)

    Google Scholar 

  25. Fornberg, B.: A pseudospectral fictitious point method for high order initial-boundary value problems. SIAM J. Sci. Comput. 28(5), 1716–1729 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Fornberg, B., Sloan, C.M.: A review of pseudospectral methods for solving partial differential equations. Acta Numerica 3, 203–267 (1994)

    Article  MathSciNet  Google Scholar 

  27. Fornberg, B., Whitham, G.B.: A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. Roy. Soc. A 289, 373–404 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  28. Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE 93(2), 216–231 (2005)

    Article  Google Scholar 

  29. Galgani, L., Giorgilli, A., Martinoli, A., Vanzini, S.: On the problem of energy equipartition for large systems of the Fermi–Pasta–Ulam type: analytical and numerical estimates. Physica D 59, 334–348 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. Gallego, R., Castro, M., López, J.M.: Pseudospectral versus finite-difference schemes in the numerical integration of stochastic models of surface growth. Phys. Rev. E 76(5), 51121 (2007)

    Article  Google Scholar 

  31. Giovine, P., Oliveri, F.: Dynamics and wave propagation in dilatant granular materials. Meccanica 30(4), 341–357 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  32. Goursolle, T., Callé, S., Dos Santos, S., Matar, O.B.: A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy. J. Acoust. Soc. Am. 122, 3220 (2007)

    Article  Google Scholar 

  33. Guo, J., Taha, T.R.: Parallel implementation of the split-step and the pseudospectral methods for solving higher KdV equation. Math. Comput. Simulat. 62(1-2), 41–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Guyenne, P., Nicholls, D.P.: A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM J. Sci. Comput. 30(1), 81–101 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hindmarsh, A.C.: ODEPACK, a systematized collection of ODE solvers. In: Stepleman, R.S., et al. (eds.) Scientific Computing: Applications of Mathematics and Computing to the Physical Sciences, IMACS Trans. Sci. Comput., Vol. 1, pp. 55–64. North-Holland, Amsterdam (1983)

    Google Scholar 

  36. Ilison, L., Salupere, A.: On solitons in dilatant granular materials. In: Lund, E., Olhoff, N., Stegman, J. (eds.) Proceedings 15th Nordic Seminar on Computational Mechanics 2002, pp. 181–184. Institute of Mechanical Enginnering, Aalborg University (2002)

    Google Scholar 

  37. Ilison, L., Salupere, A.: Solitons in hierarchical Korteweg–de Vries type systems. Proc. Estonian Acad. Sci. Phys. Math. 52(1), 135–144 (2003)

    MATH  MathSciNet  Google Scholar 

  38. Ilison, L., Salupere, A.: Propagation of localised perturbations in granular materials. Research Report Mech 287/07, Institute of Cybernetics at Tallinn University of Technology (2007)

    Google Scholar 

  39. Ilison, L., Salupere, A.: Interactions of solitary waves in hierarchical KdV-type system. Research Report Mech 291/08, Institute of Cybernetics at Tallinn University of Technology (2008)

    Google Scholar 

  40. Ilison, L., Salupere, A.: Propagation of sech2-type solitary waves in hierarchical KdV-type systems. Math. Comput. Simulat. 79, 3314–3327 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ilison, L., Salupere, A.: Numerical simulation of interaction of solitons and solitary waves in hierarchical KdV-type systems. Commun. Nonlinear Sci. Numer. Simulat. (2009) (submitted)

    Google Scholar 

  42. Ilison, L., Salupere, A., Peterson, P.: On the propagation of localised perturbations in media with microstructure. Proc. Estonian Acad. Sci. Phys. Math. 56(2), 84–92 (2007)

    MATH  MathSciNet  Google Scholar 

  43. Ilison, O., Salupere, A. On propagation of solitons in media with higher order dispersion. In: Lund, E., Olhoff, N., Stegman, J. (eds.) Proceedings 15th Nordic Seminar on Computational Mechanics 2002, pp. 177–180. Institute of Mechanical Enginnering, Aalborg University (2002)

    Google Scholar 

  44. Ilison, O., Salupere, A.: On the formation of solitons in media with higher-order dispersive effects. Proc. Estonian Acad. Sci. Phys. Math. 52(1), 135–144 (2003)

    MATH  MathSciNet  Google Scholar 

  45. Ilison, O., Salupere, A.: On the propagation of solitary waves in microstructured solids. In: Gutkowski, W., Kowalewski, T.A. (eds.) 21st International Congress of Theoretical and Applied Mechanics 2004, ICTAM04 CD-ROM Proceedings. IPPT PAN, Warszawa (2004)

    Google Scholar 

  46. Ilison, O., Salupere, A.: Propagation of sech2-type solitary waves in higher order KdV-type systems. Chaos Soliton.Fract. 26, 453–465 (2005)

    Article  MATH  Google Scholar 

  47. Ilison, O., Salupere, A.: On the propagation of solitary pulses in microstructured materials. Chaos Soliton. Fract. 29, 202–214 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  48. Janno, J., Engelbrecht, J.: An inverse solitary wave problem related to microstructured materials. Inverse Problems 21, 2019–2034 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  49. Janno, J., Engelbrecht, J.: Solitary waves in nonlinear microstructured materials. J. Phys. A: Math. Gen. 38, 5159–5172 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  50. Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2007). http://www.scipy.org

  51. Keetels, G.H., D’Ortona, U., Kramer, W., Clercx, H.J.H., Schneider, K., Heijst, van G.J.F.: Fourier spectral and wavelet solvers for the incompressible Navier–Stokes equations with volume-penalization: Convergence of a dipole–wall collision. J. Comput. Phys. 227(2), 919–945 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  52. Kliakhandler, I.L., Porubov, A.V., Velarde, M.G.: Localized finite-amplitude disturbances and selection of solitary waves. Phys. Rev. E 62(4), 4959–4962 (2000)

    Article  Google Scholar 

  53. Kreiss, H.O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199–215 (1972)

    Article  MathSciNet  Google Scholar 

  54. Lee, J.: Free vibration analysis of cylindrical helical springs by the pseudospectral method. J. Sound Vibration 302(1-2), 185–196 (2007)

    Article  Google Scholar 

  55. Lee, J.: Free vibration analysis of non-cylindrical helical springs by the pseudospectral method. J. Sound Vibration 305(3), 543–551 (2007)

    Article  Google Scholar 

  56. Lo, J., Shizgal, B.D.: Spectral convergence of the quadrature discretization method in the solution of the Schrödinger and Fokker–Planck equations: Comparison with sinc methods. J. Chem. Phys. 125, 194108 (2006)

    Article  Google Scholar 

  57. Mead, J., Zubik-Kowal, B.: An iterated pseudospectral method for delay partial differential equations. Appl. Numer. Math. 55(2), 227–250 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  58. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  59. Miura, R.M.: The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18(3), 412–459 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  60. Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations II. Existence of conservation laws and constants of motion. J. Math. Phys. 9, 1204–1209 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  61. Olmos, D., Shizgal, B.D.: A pseudospectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 193(1), 219–242 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  62. Oran Brigham, E.: The Fast Fourier Transform and Its Applications. Prentice Hall, London (1988)

    Google Scholar 

  63. Orszag, S.A.: Comparison of pseudospectral and spectral approximation. Stud. Appl. Math. 51, 253–259 (1972)

    MATH  Google Scholar 

  64. Pelinovsky, E., Sergeeva, A.: Numerical modeling of the KdV random wave field. Eur. J. Mech. B Fluids 25(4), 425–434 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  65. Peterson, P.: F2PY: Fortran to Python interface generator (2005). http://cens.ioc.ee/projects/f2py2e/

  66. Peterson, P., Salupere, A.: Solitons in a perturbed Korteweg-de Vries system. Proc. Estonian Acad. Sci. Phys. Math. 46(1-2), 102–110 (1997)

    MATH  Google Scholar 

  67. Porubov, A.V., Maugin, G.A., Gursky, V.V., Krzhizhanovskaya, V.V.: On some localized waves described by the extended KdV equation. C.R. Mecanique 333(7), 528–533 (2005)

    Google Scholar 

  68. Rashid, A.: Convergence analysis of three-level Fourier pseudospectral method for Korteweg–de Vries Burgers equation. Comput. Math. Appl. 52(5), 769–778 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  69. Salupere, A.: On the application of the pseudospectral method for solving the Korteweg-de Vries equation. Proc. Estonian Acad. Sci. Phys. Math. 44(1), 73–87 (1995)

    MATH  MathSciNet  Google Scholar 

  70. Salupere, A.: On the application of pseudospectral methods for solving nonlinear evolution equations, and discrete spectral analysis. In: Proceedings of 10th Nordic Seminar on Computational Mechanics, pp. 76–83. Tallinn Technical University, Tallinn (1997)

    Google Scholar 

  71. Salupere, A.: Technique for detection of solitons in numerical experiments and virtual soliton concept. In: Oñate, E., Bugeda, G., Suárez, G. (eds.) CD-ROM Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering – ECCOMAS 2000, Barcelona (2000).

    Google Scholar 

  72. Salupere, A., Engelbrecht, J. Hidden and driven solitons in microstructured media. In: Gutkowski, W. Kowalewski, T.A. (eds.) 21st International Congress of Theoretical and Applied Mechanics ICTAM04 CD-ROM Proceedings. IPPT PAN, Warszawa (2004)

    Google Scholar 

  73. Salupere, A., Engelbrecht, J., Ilison, O., Ilison, L.: On solitons in microstructured solids and granular materials. Math. Comput. Simulat. 69, 502–513 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  74. Salupere, A., Engelbrecht, J., Maugin, G.A.: Solitonic structures in KdV-based higher-order systems. Wave Motion 34, 51–61 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  75. Salupere, A., Engelbrecht, J., Peterson, P.: Long-time behaviour of soliton ensembles. Part I–Emergence of ensembles. Chaos Soliton. Fract. 14, 1413–1424 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  76. Salupere, A., Engelbrecht, J., Peterson, P.: Long-time behaviour of soliton ensembles. Part II–Periodical patterns of trajectories. Chaos Soliton. Fract. 15, 29–40 (2003)

    Article  MathSciNet  Google Scholar 

  77. Salupere, A., Engelbrecht, J., Peterson, P.: On the long-time behaviour of soliton ensembles. Math. Comput. Simulat. 62, 137–147 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  78. Salupere, A., Ilison, O.: On the numerical determination of solitary waves for systems with quartic potential and higher order dispersion. In: Eriksson, A., Pacoste, C. (eds.) Proceedings of the NSCM-11: Nordic Seminar on Computational Mechanics. TRITA-BKN. Bulletin 39, pp. 106–109. KTH, Stockholm (1998)

    Google Scholar 

  79. Salupere, A., Ilison, O.: On solitonic structures in microstructured materials. In: Proceedings 13th Nordic Seminar on Computational Mechanics (NSCM-13) 2000, pp. 70–73. Oslo (2000)

    Google Scholar 

  80. Salupere, A., Kukk, M.: Periodically forced solitonic structures in dispersive media. Proc. Estonian Acad. Sci. Phys. Math. 52(1), 145–156 (2003)

    MATH  MathSciNet  Google Scholar 

  81. Salupere, A., Maugin, G.A., Engelbrecht, J.: Korteweg–de Vries soliton detection from a harmonic input. Phys. Lett. A 192(1), 5–8 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  82. Salupere, A., Maugin, G.A., Engelbrecht, J.: Solitons in systems with a quartic potential and higher-order dispersion. Proc. Estonian Acad. Sci. Phys. Math. 46(1–2), 118–127 (1997)

    MATH  MathSciNet  Google Scholar 

  83. Salupere, A., Maugin, G.A., Engelbrecht, J., Kalda, J.: On the KdV soliton formation and discrete spectral analysis. Wave Motion 23(1), 49–66 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  84. Salupere, A., Randrüüt, M., Tamm, K.: Emergence of soliton trains in microstructured materials. In: Denier, J., Finn, M., Mattner, T. (eds.) XXII International Congress of Theoretical and Applied Mechanics ICTAM 2008, CD-ROM Proceedings. The University of Adelaide, Adelaide (2008). Paper id 11311

    Google Scholar 

  85. Salupere, A., Tamm, K., Engelbrecht, J.: Numerical simulation of interaction of solitary deformation waves in microstructured solids. Int. J. Non-linear Mech. 43, 201–208 (2008)

    Article  Google Scholar 

  86. Salupere, A., Tamm, K., Engelbrecht, J., Peterson, P.: On interaction of deformation waves in microstructured solids. Proc. Estonian Acad. Sci. Phys. Math. 56(2), 93–99 (2007)

    MATH  MathSciNet  Google Scholar 

  87. Sarra, S.A.: A Pseudospectral Method with Edge Detection-Free Postprocessing for Two-Dimensional Hyperbolic Heat Transfer. Numer. Heat Transfer, Part B 54(1), 52–61 (2008)

    Article  MathSciNet  Google Scholar 

  88. Vallikivi, M., Salupere, A., Dai, H.H.: Numerical simulation of propagation of solitary deformation waves in a compressible hyperelastic rod. Research Report Mech 293/08, Institute of Cybernetics at Tallinn University of Technology (2008)

    Google Scholar 

  89. Veski, K., Engelbrecht, J., Bountis, T.: Emergence and propagation of nerve pulses: parameter analysis and response to forcing. Chaos Soliton.Fract. (2008) (submitted)

    Google Scholar 

  90. Wang, T., Guo, B.: Composite generalized Laguerre–Legendre pseudospectral method for Fokker–Planck equation in an infinite channel. Appl. Numer. Math. 58(10), 1448–1466 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  91. Yin, Z., Clercx, H.J.H., Montgomery, D.C.: An easily implemented task-based parallel scheme for the Fourier pseudospectral solver applied to 2D Navier–Stokes turbulence. Comput. Fluids 33(4), 509–520 (2004)

    Article  MATH  Google Scholar 

  92. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618, Tallinn, Estonia

    Andrus Salupere

Authors
  1. Andrus Salupere
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrus Salupere .

Editor information

Editors and Affiliations

  1. Tallinn University of Technology, Inst. Cybernetics, Estonian Academy of Sciences, Akadeemia Tee 21, Tallinn, 12618, Estonia

    Ewald Quak

  2. Tallinn University of Technology, Inst. Cybernetics, Estonian Academy of Sciences, Akadeemia Tee 21, Tallinn, 12618, Estonia

    Tarmo Soomere

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Salupere, A. (2009). The Pseudospectral Method and Discrete Spectral Analysis. In: Quak, E., Soomere, T. (eds) Applied Wave Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00585-5_16

Download citation

Publish with us

Policies and ethics

Search

Navigation