Skip to main content
SpringerLink
Log in

Difference Schemes Based on the Laguerre Transform

  • GENERAL NUMERICAL METHODS
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Optimal difference schemes based on the Laguerre transform are proposed for solving the wave equation. Additional parameters are introduced into the difference scheme used for the equations of harmonics. Numerical values of these parameters are obtained by minimizing the error in the difference approximation of the Helmholtz equation. The optimal parameter values thus obtained are used to construct optimal difference schemes. Optimal difference schemes of second- and fourth-order accuracy are considered. The optimal parameters of the difference schemes are presented. Their values depend only on the ratio of the spatial step sizes. It is shown that the use of optimal difference schemes improves the accuracy of solutions. The efficiency of the algorithm is enhanced by applying a simple modification of the difference scheme.

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.

Fig. 1.
Fig. 2.

Similar content being viewed by others

REFERENCES

  1. R. Luebbers and F. P. Hansberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).

    Article  Google Scholar 

  2. G. Turner and A. F. Siggins, “Constant Q attenuation of subsurface radar pulses,” Geophysics 59, 1192–1200 (1994).

    Article  Google Scholar 

  3. T. Bergmann, J. O. A. Robertsson, and K. Holliger, “Finite difference modeling of electromagnetic wave in dispersive and attenuating media,” Geophysics 63, 856–867 (1998).

    Article  Google Scholar 

  4. T. Bergmann, J. O. Blanch, J. O. A. Robertsson, and K. Holliger, “A simplified Lax–Wendroff correction for staggered-grid FDTD modeling of electromagnetic wave in frequency-dependent media,” Geophysics 64, 1369–1377 (1999).

    Article  Google Scholar 

  5. Electrical Prospecting: Geophysicist’s Manual, Ed. by A. G. Tarkhov (Nedra, Moscow, 1980) [in Russian].

    Google Scholar 

  6. G. V. Konyukh and B. G. Mikhailenko, “Use of the Laguerre integral transform in solving dynamic seismic problems,” Tr. Inst. Vychisl. Mat. Mat. Geofiz. Mat. Model. Geofiz. Novosibirsk, No. 5, 107–112 (1998).

  7. A. F. Mastryukov and B. G. Mikhailenko, “Propagation and attenuation of electromagnetic waves in inhomogeneous media: Numerical modeling using Laguerre transform,” Geol. Geofiz. 44 (10), 1060–1069 (2003).

    Google Scholar 

  8. A. F. Mastryukov and B. G. Mikhailenko, “Modeling of electromagnetic wave propagation in relaxation media by the spectral Laguerre transform,” Geol. Geofiz. 47 (3), 397–407 (2006).

    Google Scholar 

  9. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins Univ. Press, Baltimore, Md., 1996).

    MATH  Google Scholar 

  10. C. K. Tam and J. C. Webb, “Dispersion-relation-preserving finite difference schemes for computational acoustics,” J. Comput. Phys. 107 (2), 262–281 (1993).

    Article  MathSciNet  Google Scholar 

  11. C. H. Jo, C. Shin, and H. S. Suh, “An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator,” Geophysics 61, 529–537 (1996).

    Article  Google Scholar 

  12. J. B. Chen, “An average derivative optimal scheme for frequency-domain scalar wave equation,” Geophysics 77, T201–T210 (2012).

    Article  Google Scholar 

  13. A. F. Mastryukov and B. G. Mikhailenko, “Optimal finite difference schemes for Maxwell’s equations in direct problems of electromagnetic sounding,” Geol. Geofiz. 56 (9), 1713–1722 (2015).

    Google Scholar 

  14. A. F. Mastryukov, “Optimal finite difference schemes for a wave equation,” Numer. Anal. Appl. 9 (4), 299–311 (2016).

    Article  MathSciNet  Google Scholar 

  15. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Nauka, Moscow, 1982; Butterworth-Heinemann, Oxford, 1984).

  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

    MATH  Google Scholar 

  17. F. P. Vasil’ev, Numerical Methods for Solving Optimization Problems (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  18. M. Ghris, B. Fornberg, and T. A. Driscoll, “Staggered time integrator for wave equations,” SIAM J. Numer. Anal. 38, 718–741 (2000).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    A. F. Mastryukov

Authors
  1. A. F. Mastryukov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. F. Mastryukov.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mastryukov, A.F. Difference Schemes Based on the Laguerre Transform. Comput. Math. and Math. Phys. 61, 351–358 (2021). https://doi.org/10.1134/S0965542521030131

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542521030131

Keywords:

Search

Navigation