Physics of the Earth and Planetary Interiors 171 (2008) 112–121 Contents lists available at ScienceDirect Physics of the Earth and Planetary Interiors journal homepage: www.elsevier.com/locate/pepi Comparison of finite difference and finite element methods for simulating two-dimensional scattering of elastic waves Marcel Frehner a,∗ , Stefan M. Schmalholz a , Erik H. Saenger a,b , Holger Steeb c a Geological Institute, Department of Earth Sciences, ETH Zurich, 8092 Zurich, Switzerland b Spectraseis AG, 8005 Zurich, Switzerland c Multi Scale Mechanics, TS, CTW, University of Twente, 7500 AE Enschede, The Netherlands article info Article history: Received 30 October 2007 Received in revised form 10 June 2008 Accepted 7 July 2008 Keywords: Scattering Wave propagation Numerical methods Analytical solution Finite element method Finite difference method abstract Two-dimensional scattering of elastic waves in a medium containing a circular heterogeneity is investi- gated with an analytical solution and numerical wave propagation simulations. Different combinations of finite difference methods (FDM) and finite element methods (FEM) are used to numerically solve the elas- todynamic wave equations. Finite difference and finite element techniques are applied to approximate both the time and space derivatives and are combined in various ways to provide different numerical algorithms for modeling elastic wave propagation. The results of the different numerical algorithms are compared for simulations of an incident plane P-wave that is scattered by a mechanically weak circu- lar inclusion whereby the diameter of the inclusion is of the same order than the P-wave’s wavelength. For this scattering problem an analytical solution is available and used as the reference solution in the comparison of the different numerical algorithms. Staircase-like spatial discretization of the inclusion’s circular shape with the finite difference method using a rectangular grid provides accurate velocity and displacement fields close to the inclusion boundary only for very high spatial resolutions. Implicit time integration based on either finite differences or finite elements does not provide computational advan- tages compared to explicit schemes. The best numerical algorithm in terms of accuracy and computation time for the investigated scattering problem consists of a finite element method in space using an unstruc- tured mesh combined with an explicit finite difference method in time. The computational advantages and disadvantages of the different numerical algorithms are discussed. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Propagation of seismic waves can be described analytically for some specific geometrical setups (Love, 1927; Achenbach, 1973; Aki and Richards, 1980; Ben-Menahem and Jit Singh, 1981). For more complex geometries, ray-tracing methods (Moser and Pajchel, 1997; Cerveny, 2001) are able to approximate propagation of high- frequency seismic waves when the wavelength is significantly smaller than the characteristic size of heterogeneities. For seismic waves having a significantly larger wavelength than the character- istic size of heterogeneities, effective medium theories can be used (Mavko et al., 1998). However, if the wavelengths of the propagat- ing waves and the characteristic size of heterogeneities are of the same order, numerical methods are essential. Particular numeri- cal challenges are for example scattering phenomena in complex geometries (Korneev and Johnson, 1996), wave attenuation due to ∗ Corresponding author. Tel.: +41 44 632 88 72; fax: +41 44 632 10 30. E-mail address: marcel.frehner@erdw.ethz.ch (M. Frehner). wave induced fluid flow (Carcione et al., 2003; Masson and Pride, 2007), wave propagation in three-phase media (Carcione et al., 2004; Santos et al., 2005) or microscale modeling of wave prop- agation in poroelastic rocks (Saenger et al., 2007). Although on different scales, all these challenges comprise wave scattering at heterogeneities. For numerical modeling of seismic wave propagation different methods are available (Kelly and Marfurt, 1990; Carcione et al., 2002; Cohen, 2002) which can have advantages and disadvantages depending on the particular problem under study. Methods used in this paper are the finite difference method (FDM) (Smith, 1985; Ames, 1992; Moczo et al., 2007) and the finite element method (FEM) (Hughes, 1987; Bathe, 1996; Zienkiewicz and Taylor, 2000). Both methods can be used to discretize spatial as well as time derivatives. Different combinations of spatial and temporal dis- cretization methods using FDM and FEM are compared in this study. The different algorithms are described and applied to a two-dimensional (2D) elastic scattering problem for comparison. Analytical solutions for scattered wave fields are available for dif- ferent cases (Ying and Truell, 1956; White, 1958; Liu et al., 2000; 0031-9201/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2008.07.003