3-D finite-difference, finite-element, discontinuous-Galerkin and spectral-element schemes analysed for their accuracy with respect to P-wave to S-wave speed ratio
Article first published online: 18 OCT 2011
© 2011 The Authors Geophysical Journal International © 2011 RAS
Geophysical Journal International
Volume 187, Issue 3, pages 1645–1667, December 2011
How to Cite
Moczo, P., Kristek, J., Galis, M., Chaljub, E. and Etienne, V. (2011), 3-D finite-difference, finite-element, discontinuous-Galerkin and spectral-element schemes analysed for their accuracy with respect to P-wave to S-wave speed ratio. Geophysical Journal International, 187: 1645–1667. doi: 10.1111/j.1365-246X.2011.05221.x
- Issue published online: 17 NOV 2011
- Article first published online: 18 OCT 2011
- Accepted 2011 September 5. Received 2011 August 15; in original form 2011 April 19
- Numerical approximations and analysis;
- Computational seismology;
- Theoretical seismology
We analyse 13 3-D numerical time-domain explicit schemes for modelling seismic wave propagation and earthquake motion for their behaviour with a varying P-wave to S-wave speed ratio (VP/VS). The second-order schemes include three finite-difference, three finite-element and one discontinuous-Galerkin schemes. The fourth-order schemes include three finite-difference and two spectral-element schemes. All schemes are second-order in time. We assume a uniform cubic grid/mesh and present all schemes in a unified form. We assume plane S-wave propagation in an unbounded homogeneous isotropic elastic medium. We define relative local errors of the schemes in amplitude and the vector difference in one time step and normalize them for a unit time. We also define the equivalent spatial sampling ratio as a ratio at which the maximum relative error is equal to the reference maximum error. We present results of the extensive numerical analysis.
We theoretically (i) show how a numerical scheme sees the P and S waves if the VP/VS ratio increases, (ii) show the structure of the errors in amplitude and the vector difference and (iii) compare the schemes in terms of the truncation errors of the discrete approximations to the second mixed and non-mixed spatial derivatives.
We find that four of the tested schemes have errors in amplitude almost independent on the VP/VS ratio.
The homogeneity of the approximations to the second mixed and non-mixed spatial derivatives in terms of the coefficients of the leading terms of their truncation errors as well as the absolute values of the coefficients are key factors for the behaviour of the schemes with increasing VP/VS ratio.
The dependence of the errors in the vector difference on the VP/VS ratio should be accounted for by a proper (sufficiently dense) spatial sampling.