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Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture

Day, Steven M. and Dalguer, Luis A. and Lapusta, Nadia and Liu, Yi (2005) Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture. Journal of Geophysical Research B, 110 (B12). Art. No. B12307. ISSN 0148-0227. doi:10.1029/2005JB003813.

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The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Δx is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Δx, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.

Item Type:Article
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Lapusta, Nadia0000-0001-6558-0323
Additional Information:© 2005 American Geophysical Union. Received 2 May 2005; revised 5 October 2005; accepted 12 October 2005; published 23 December 2005. The authors thank Ruth Harris and Ralph Archuleta for organizing the Southern California Earthquake Center (SCEC) Spontaneous Rupture Code-Validation project, which provided the initial impetus to compare the BI and DFM methods. Jean Paul Ampuero, Peter Moczo, and Joe Andrews provided very helpful reviews, leading to improvements in the manuscript. Jean Virieux pointed out connections between the DFM scheme and other finite difference methods. This work was supported by the National Science Foundation, under grants ATM-0325033 and EAR-0122464 (SCEC Community Modeling Environment Project), and by SCEC. SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. This is SCEC contribution 907.
Funding AgencyGrant Number
Southern California Earthquake Center (SCEC)UNSPECIFIED
Subject Keywords:finite difference; boundary integral; earthquake simulation
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Other Numbering System NameOther Numbering System ID
Southern California Earthquake Center (SCEC)907
Issue or Number:B12
Record Number:CaltechAUTHORS:20130416-143417195
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Official Citation:Day, S. M., L. A. Dalguer, N. Lapusta, and Y. Liu (2005), Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture, J. Geophys. Res., 110, B12307, doi:10.1029/2005JB003813
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:37974
Deposited By: Tony Diaz
Deposited On:17 Apr 2013 16:42
Last Modified:09 Nov 2021 23:32

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