## 1. Introduction

[2] The availability of high-quality near-field records of large subduction earthquakes in the last few years makes it possible to test and validate physics-based rupture models. The development of sophisticated models to explain such aggregate of observations is now largely justified. Huge efforts have been made by the seismological community in the last ten years to overcome technical limitations preventing most methods for dynamic rupture calculations from integrating the effect of fault geometry in the spontaneous rupture of earthquakes. Because of both its simplicity and efficiency, the finite difference (FD) method has been one of the first and most persistent approaches to simulate rupture dynamics along planar faults [e.g.,*Andrews*, 1976; *Madariaga*, 1976; *Miyatake*, 1980; *Day*, 1982; *Virieux and Madariaga*, 1982; *Harris and Day*, 1993; *Madariaga et al.*, 1998; *Peyrat et al.*, 2001; *Day et al.*, 2005; *Dalguer and Day*, 2007]. Although different strategies have been proposed in recent years to integrate complex fault geometries into such methods [*Cruz-Atienza and Virieux*, 2004; *Kase and Day*, 2006; *Cruz-Atienza et al.*, 2007; *Kozdon et al.*, 2012], most common approaches handle numerical lattices (meshes) that are adaptable to the problem geometry (i.e., fault geometry). One set of methods is based on well established boundary integral equations (BIE), [e.g., *Das and Aki*, 1977; *Andrews*, 1985; *Cochard and Madariaga*, 1994; *Kame and Yamashita*, 1999; *Aochi et al.*, 2000; *Lapusta et al.*, 2000; *Hok and Fukuyama*, 2011]. However, since these methods discretize only boundaries and require semi-analytical approximations of Green functions, they have difficulties integrating heterogeneities of the bulk properties into which the fault is embedded. The other set consists of domain methods based on weak formulations of the elastodynamic equations, and can be separated into two subgroups depending on how the lattice boundaries are treated. On one hand the continuous finite element methods (FEM), whose formulations require continuity between the mesh elements except where special treatments of boundary conditions are imposed [e.g.,*Oglesby and Day*, 2001; *Ampuero*, 2002; *Festa and Vilotte*, 2006; *Ma and Archuleta*, 2006; *Kaneko et al.*, 2008; *Ely et al.*, 2009; *Barall*, 2009]. On the other, the discontinuous finite element methods, better known as the discontinuous Galerkin (DG) methods, which only consider fluxes between elements and, therefore, do not impose any field continuity across their boundaries.

[3] When studying the earthquakes source physics, the discontinuity produced across the fault by the rupture process must be accurately treated, so that the DG strategy is naturally suitable for tackling this problem.

[4] The first dynamic rupture model based on a DG approach was introduced in two dimensions (2D) by *Benjemaa et al.* [2007]for low-order (P0) interpolation functions. In this case, where the basis functions are constants, the DG schemes are also known as finite volume (FV) methods [*LeVeque*, 2002] and provide computationally efficient algorithms that are as fast as second order FD schemes (i.e., they are equivalent in efficiency on rectangular meshes). However, the extension to three dimensions (3D) of this model [*Benjemaa et al.*, 2009] revealed convergence problems for unstructured tetrahedral grids (e.g., non-planar faults) [*Tago et al.*, 2010]. On these irregular grids, P0 elements have zero-order convergence for wave propagation modeling due to the centered flux approximation [*Brossier et al.*, 2009; *Remaki et al.*, 2011], so increasing the element interpolation order to achieve a proper numerical convergence of wave propagation with a DG scheme is mandatory. Nevertheless, in practice, high-order convergence rates are not clearly observed for the dynamic-rupture numerical problem (i.e., 4th order or higher), and second order interpolation methods are often the most accurate and efficient approximations for applying the corresponding fault boundary conditions [*Cruz-Atienza et al.*, 2007; *Moczo et al.*, 2007; *Rojas et al.*, 2009; *Kozdon et al.*, 2012]. A notable case for which the convergence rate is essentially insensitive to increments in the interpolation order is the ADER-DG discontinuous Galerkin method for 2D and 3D geometries by*de la Puente et al.* [2009] and *Pelties et al.* [2012], respectively, despite its spectral convergence for the wave propagation problem [*Dumbser and Käser*, 2006]. The ADER-DG is based on a modal interpolation formulation, instead of the nodal interpolation we consider here. Both formulations are mathematically equivalent but computationally different [*Hesthaven and Warburton*, 2008]. Our choice of using the nodal approximation essentially relies on the fact that the evaluation of fluxes requires fewer computations than in a modal scheme, as we shall explain on section 4.1.

[5] In this work we introduce a novel discontinuous Galerkin approach, namely the DGCrack method, to model the dynamic rupture of earthquakes in 3D geometries. The numerical platform of our model is the GeoDG3D parallel code [*Etienne et al.*, 2010] developed for the elastic wave propagation. For the parallel implementation it uses the Message Passing Interface (MPI) and achieves ∼80% strong scalability. GeoDG3D accounts for free surface boundary conditions along arbitrary topographies, and includes Convolutional Perfectly Matching Layer (CPML) absorbing boundary conditions at the external edges of the physical domain [*Etienne et al.*, 2010, and references therein]. Furthermore, intrinsic attenuation has been recently introduced into GeoDG3D via the rock quality *Q* [*Tago et al.*, 2010], but will not be discussed in the present work. To maximize both the efficiency and the accuracy of the scheme depending on the model properties and geometry, the method handles unstructured mesh refinements (i.e., h-adaptivity) and locally adapts the order of the nodal interpolations (i.e., within every grid element; p-adaptivity) [*Etienne et al.*, 2010].

[6] We first introduce the mathematical and computational concepts for the 3D dynamic rupture problem, and then assess both its accuracy and convergence rate by comparing calculated solutions with those yielded by finite difference (DFM), finite element (FEM), spectral boundary integral (MDSBI) and spectral element (SPECFEM3D) methods for two spontaneous rupture benchmark tests [*Harris et al.*, 2009]. Since one of our major goals in the near future is the investigation of dynamic rupture propagation along realistic (nonplanar) fault geometries, we take special care to verify the accuracy of the normal stress field across the fault during rupture propagation, as the fault normal tractions strongly determine the radiated energy throughout the Coulomb failure criterion. We finally illustrate the capabilities of the DGCrack method through a spontaneous rupture simulation along the 1992 Landers earthquake fault system, which is a geometrically intricate and physically interesting study case.