S U M M A R YWe present a new numerical method to solve the heterogeneous anelastic, seismic wave equations with arbitrary high order accuracy in space and time on 3-D unstructured tetrahedral meshes. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method combines the Discontinuous Galerkin (DG) finite element (FE) method with the ADER approach using Arbitrary high order DERivatives for flux calculations. The DG approach, in contrast to classical FE methods, uses a piecewise polynomial approximation of the numerical solution which allows for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann problems can be applied as in the finite volume framework. The main idea of the ADER time integration approach is a Taylor expansion in time in which all time derivatives are replaced by space derivatives using the so-called Cauchy-Kovalewski procedure which makes extensive use of the governing PDE. Due to the ADER time integration technique the same approximation order in space and time is achieved automatically and the method is a one-step scheme advancing the solution for one time step without intermediate stages. To this end, we introduce a new unrolled recursive algorithm for efficiently computing the Cauchy-Kovalewski procedure by making use of the sparsity of the system matrices. The numerical convergence analysis demonstrates that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes while computational cost and storage space for a desired accuracy can be reduced when applying higher degree approximation polynomials. In addition, we investigate the increase in computing time, when the number of relaxation mechanisms due to the generalized Maxwell body are increased. An application to a well-acknowledged test case and comparisons with analytic and reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-DG approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

[1] Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic data into emerging approaches for dynamic source inversion, and to generate realistic physics-based earthquake scenarios for hazard assessment. Modeling of spontaneous earthquake rupture and seismic wave propagation by a high-order discontinuous Galerkin (DG) method combined with an arbitrarily high-order derivatives (ADER) time integration method was introduced in two dimensions by de la Puente et al. (2009). The ADER-DG method enables high accuracy in space and time and discretization by unstructured meshes. Here we extend this method to three-dimensional dynamic rupture problems. The high geometrical flexibility provided by the usage of tetrahedral elements and the lack of spurious mesh reflections in the ADER-DG method allows the refinement of the mesh close to the fault to model the rupture dynamics adequately while concentrating computational resources only where needed. Moreover, ADER-DG does not generate spurious high-frequency perturbations on the fault and hence does not require artificial Kelvin-Voigt damping. We verify our three-dimensional implementation by comparing results of the SCEC TPV3 test problem with two well-established numerical methods, finite differences, and spectral boundary integral. Furthermore, a convergence study is presented to demonstrate the systematic consistency of the method. To illustrate the capabilities of the high-order accurate ADER-DG scheme on unstructured meshes, we simulate an earthquake scenario, inspired by the 1992 Landers earthquake, that includes curved faults, fault branches, and surface topography.Citation: Pelties, C., J. de la Puente, J.-P. Ampuero, G. B. Brietzke, and M. Käser (2012), Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes,

We have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded threedimensional domains. This method is a discontinuous Galerkin method that achieves arbitrary high-order accuracy on unstructured tetrahedral meshes for the low-frequency range and the inviscid case. By using Biot's equations and Darcy's dynamic laws, we have built a scheme that can successfully model wave propagation in fluid-saturated porous media when anisotropy of the pore structure is allowed. Zero-inflow fluxes are used as absorbing boundary conditions. A continuous arbitrary high-order derivatives time integration is used for the high-frequency inviscid case, whereas a space-time discontinuous scheme is applied for the low-frequency case. We conducted a numerical convergence test of the proposed methods. We used a series of examples to quantify the quality of our numerical results, comparing them to analytic solutions as well as numerical solutions obtained by other methodologies. In particular, a large scale 3D reservoir model showed the method's suitability to solve poroelastic wavepropagation problems for complex geometries using unstructured tetrahedral meshes. The resulting method is proved to be high-order accurate in space and time, stable for the lowfrequency case, and asymptotically consistent with the diffusion limit.

S U M M A R YWe present a new numerical method to solve the heterogeneous elastic anisotropic wave equation with arbitrary high-order accuracy in space and time on unstructured tetrahedral meshes. Using the most general Hooke's tensor we derive the velocity-stress formulation leading to a linear hyperbolic system which accounts for the variation of the material properties depending on direction. This approach allows for the accurate modelling even of the most general crystalline symmetry class, the triclinic anisotropy, as no interpolation of material properties to particular mesh vertices is necessary. The proposed method combines the Discontinuous Galerkin method with the arbitrary high-order derivatives (ADER) time integration approach using arbitrary high-order derivatives of the piecewise polynomial representation of the unknown solution. The discontinuities of this piecewise polynomial approximation at element interfaces permit the application of the well-established theory of finite volumes and numerical fluxes across element interfaces obtained by the solution of derivative Riemann problems. Due to the novel ADER time integration technique the scheme provides the same approximation order in space and time automatically. A numerical convergence study confirms that the new scheme achieves the desired arbitrary high-order accuracy even for anisotropic material on unstructured tetrahedral meshes. Furthermore, it shows that higher accuracy can be reached with higher-order schemes while reducing computational cost and storage space. To this end, we also present a new Godunov-type numerical flux for anisotropic material and compare its accuracy with a computationally simpler Rusanov flux. As a further extension, we include the coupling of anisotropy and viscoelastic attenuation based on the Generalized Maxwell Body rheology and the mean and deviatoric stress concepts. Finally, we validate the new scheme by comparing the results of our simulations to an analytic solution as well as to spectral element computations.

[1] We introduce the application of an arbitrary high-order derivative (ADER) discontinuous Galerkin (DG) method to simulate earthquake rupture dynamics. The ADER-DG method uses triangles as computational cells which simplifies the process of discretization of very complex surfaces and volumes by using external automated tools. Discontinuous Galerkin methods are well suited for solving dynamic rupture problems in the velocity-stress formulation as the variables are naturally discontinuous at the interface between two elements. Therefore, the fault has to be honored by the computational mesh. The so-called Riemann problem can be solved to obtain well defined values of the variables at the discontinuity itself. Fault geometries of high complexity can be modeled thanks to the flexibility of unstructured meshes, which solves a major bottleneck of other high-order numerical methods. Additionally, element refinement and coarsening are easily controlled in the meshing process to better resolve the near-fault area of the model. The fundamental properties of the method are shown, as well as a series of validating exercises with reference solutions and a comparison with the well-established finite difference, boundary integral, and spectral element methods, in order to test the accuracy of our formulation. An example of dynamic rupture on a nonplanar fault based upon the Landers 1992 earthquake fault system is presented to illustrate the main potentials of the new method.Citation: de la Puente, J., J.-P. Ampuero, and M. Käser (2009), Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method,

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S U M M A R YWe present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method relies on the finite volume (FV) approach where cell-averaged quantities are evolved in time by computing numerical fluxes at the element interfaces. The basic ingredient of the numerical flux function is the solution of Generalized Riemann Problems at the element interfaces according to the arbitrary high-order derivatives (ADER) approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original first-order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniformly high order of accuracy in space and time. The high-order polynomials in space, needed as input for the numerical flux function, are obtained using a reconstruction operator acting on the cell averages. This reconstruction operator uses some techniques originally developed in the Discontinuous Galerkin (DG) Finite Element framework, namely hierarchical orthogonal basis functions in a reference element. In particular, in this article we pay special attention to underline the differences as well as the points in common with the ADER-DG schemes previously developed by the authors, especially concerning the MPI parallelization of both methods.The numerical convergence analysis demonstrates that the proposed FV schemes provide very high order of accuracy even on unstructured tetrahedral meshes while computational cost for a desired accuracy can be reduced when applying higher order reconstructions. Applications to a series of well-acknowledged elastic and anelastic test cases and comparisons with analytic and numerical reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-FV approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.