@article {DelaPuente2007,
	title = {{An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - IV. Anisotropy}},
	journal = {Geophysical Journal International},
	volume = {169},
	number = {3},
	year = {2007},
	pages = {1210{\textendash}1228},
	abstract = {We 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{\textquoteright}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. {\textcopyright} 2007 The Authors Journal compilation {\textcopyright} 2007 RAS.},
	keywords = {ADER approach, Anisotropy, Discontinuous Galerkin method, High-order accuracy in space and time, Unstructured tetrahedral meshes},
	issn = {0956540X},
	doi = {10.1111/j.1365-246X.2007.03381.x},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-34250166175\&partnerID=tZOtx3y1},
	author = {de la Puente, Josep and K{\"a}ser, Martin and Dumbser, Michael and Igel, Heiner}
}
@article {Kaser2007,
	title = {{An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - III. Viscoelastic attenuation}},
	journal = {Geophysical Journal International},
	volume = {168},
	number = {1},
	year = {2007},
	month = {jan},
	pages = {224{\textendash}242},
	abstract = {We 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. {\textcopyright} 2006 The Authors Journal compilation {\textcopyright} 2006 RAS.},
	keywords = {Attenuation, Discontinuous Galerkin, High order accuracy, Relaxation, Unstructured meshes, Viscoelasticity},
	issn = {0956540X},
	doi = {10.1111/j.1365-246X.2006.03193.x},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-33845968162\&partnerID=tZOtx3y1},
	author = {K{\"a}ser, Martin and Dumbser, Michael and de la Puente, Josep and Igel, Heiner}
}
