Existence of a second island of stability of predictor–corrector schemes for calculating synthetic seismograms

  1. Robert J. Geller1,
  2. Hiromitsu Mizutani1,2,*,
  3. Nobuyasu Hirabayashi3

Article first published online: 21 NOV 2011

DOI: 10.1111/j.1365-246X.2011.05251.x

Issue

Geophysical Journal International

Geophysical Journal International

Volume 188, Issue 1, pages 253–262, January 2012

Additional Information

How to Cite

Geller, R. J., Mizutani, H. and Hirabayashi, N. (2012), Existence of a second island of stability of predictor–corrector schemes for calculating synthetic seismograms. Geophysical Journal International, 188: 253–262. doi: 10.1111/j.1365-246X.2011.05251.x

Author Information

  1. 1

    Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan. E-mail: bob@eps.s.u-tokyo.ac.jp

  2. 2

    International Institute of Seismology and Earthquake Engineering, Building Research Institute, 1 Tatehara, Tsukuba, Ibaraki 305-0802, Japan

  3. 3

    Schlumberger K.K., 2-2-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-0206, Japan

* Now at: Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo.

Publication History

  1. Issue published online: 20 DEC 2011
  2. Article first published online: 21 NOV 2011
  3. Accepted 2011 September 27. Received 2011 September 26; in original form 2010 October 14

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Keywords:

  • Numerical solutions;
  • Numerical approximations and analysis;
  • Computational seismology;
  • Wave propagation

SUMMARY

As the first step towards a general analysis of the stability of optimally accurate predictor–corrector (P–C) time domain discretized schemes for solving the elastic equation of motion, we analyze the stability of two P–C schemes for a 1-D homogeneous case. Letting Δt be the time step, h be the spatial grid interval, β be the velocity of seismic wave propagation and inline image be the dimensionless Courant parameter, we find that each scheme has the following stability properties: stability for inline image, instability for inline image, stability for inline image and instability for inline image, where inline image. We refer to the region inline image as the second island of stability. The values of inline image, inline image and inline image are scheme-dependent. The existence of a second island of stability in a numerical scheme for solving the wave equation has not, to our knowledge, been previously reported.

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