The spectral-element method in seismology

Abstract

We present the main properties of the spectral-element method, which is well suited for numerical calculations of synthetic seismograms for three-dimensional Earth models. The technique is based upon a weak formulation of the equations of motion and combines the flexibility of a finite-element method with the accuracy of a pseudospectral method. The mesh is composed of hexahedral elements and honors the main discontinuities in the Earth model. The displacement vector is expressed in each element in terms of high-degree Lagrange interpolants, and integrals are computed based upon Gauss-Lobatto-Legendre quadrature, which leads to an exactly diagonal mass matrix and therefore drastically simplifies the algorithm. We use a fluid-solid coupling formulation that does not require iterations at the core-mantle or inner-core boundaries. The method is efficiently implemented on parallel computers with distributed memory based upon a message-passing methodology. We present two large-scale simulations for a realistic three-dimensional Earth model computed on the Japanese Earth Simulator at periods of 5 s and longer.

Spectral-Element Simulations of Seismic Waves Generated by the 2009 L'Aquila Earthquake

“…The SEM is based on a weak formulation of the seismic-wave equation that guarantees very accurate modeling of surface waves (e.g., Komatitsch et al, 2005). Moreover, the method uses high-order Lagrange polynomials to interpolate the wavefield, together with Gauss-Lobatto-Legendre (GLL) quadrature, both defined on the same GLL points, thus leading to a perfectly diagonal mass matrix.…”

“…In addition, the highdegree Lagrange polynomials assure exponential spatial accuracy. Thorough reviews of the SEM in seismology can be found, e.g., in Komatitsch et al (2005), Chaljub et al (2007), Tromp et al (2008), andFichtner (2010).…”

“…If need be, the SEM can be implemented as a discontinuous Galerkin method (Kopriva et al, 2002;Kopriva, 2006;Acosta Minolia and Kopriva, 2011). The SEM has been used to study 2D and 3D local, regional, and global seismic-wave propagation problems (e.g., Cohen et al, 1993;Priolo et al, 1994;Faccioli et al, 1997;Komatitsch, 1997;Komatitsch and Vilotte, 1998;Komatitsch and Tromp, 2002a,b;Chaljub and Valette, 2004;Komatitsch et al, 2004Komatitsch et al, , 2005Liu et al, 2004;Stich and Morelli, 2007;Lee et al, 2008;Stich et al, 2009;Stupazzini et al, 2009;Tape et al, 2010;Zhu et al, 2012a,b). In particular, in the recent work of Smerzini and Villani (2012), the SEM implemented in the code GeoELSE (Stupazzini et al, 2009) has been used to simulate near-fault ground motions produced by the 2009 L'Aquila mainshock.…”

Effects of Topography on Seismic-Wave Propagation: An Example from Northern Taiwan

“…The method has been subsequently applied in many areas of seismology (e.g., Komatitsch et al, 2002;Chaljub et al, 2003;Komatitsch et al, 2004). A critical issue for the successful application of SEM lies in the design of a good mesh (e.g., Komatitsch et al, 2005). Lee et al (2008) introduced a new mesh implementation using so-called control layers to improve mesh quality and the related numerical stability of the explicit time integration scheme.…”

Effects of Realistic Surface Topography on Seismic Ground Motion in the Yangminshan Region of Taiwan Based Upon the Spectral-Element Method and LiDAR DTM

“…The method has been subsequently applied in many areas of seismology (see, e.g., Komatitsch et al [2002], [2004] and Chaljub et al [2003]). The biggest challenge for the successful application of the SEM in the case of complex geological structures lies in the design of the mesh (Komatitsch et al, 2005). Lee, Chen, Liu, et al (2008) presented a new mesh implementation to improve mesh quality and related numerical stability.…”