@article {DelaPuente2008,
	title = {{Discontinuous Galerkin methods for wave propagation in poroelastic media}},
	journal = {GEOPHYSICS},
	volume = {73},
	number = {5},
	year = {2008},
	month = {sep},
	pages = {T77{\textendash}T97},
	abstract = {We have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three-dimensional 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{\textquoteright}s equations and Darcy{\textquoteright}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{\textquoteright}s suitability to solve poroelastic wave-propagation 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 low-frequency case, and asymptotically consistent with the diffusion limit. {\textcopyright} 2008 Society of Exploration Geophysicists. All rights reserved.},
	issn = {0016-8033},
	doi = {10.1190/1.2965027},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-53849111940\&partnerID=tZOtx3y1},
	author = {de la Puente, Josep and Dumbser, Michael and K{\"a}ser, Martin and Igel, Heiner}
}
@article {Kaser2008,
	title = {{Quantitative accuracy analysis of the discontinuous Galerkin method for seismic wave propagation}},
	journal = {Geophysical Journal International},
	volume = {173},
	number = {3},
	year = {2008},
	month = {jun},
	pages = {990{\textendash}999},
	abstract = {We present a quantitative accuracy analysis of the Discontinuous Galerkin Finite-Element method for the simulation of seismic wave propagation on tetrahedral meshes. Several parameters are responsible for the accuracy of results, such as the chosen approximation order, the spatial discretization, that is, number of elements per wavelength, and the propagation distance of the waves due to numerical dispersion and dissipation. As error norm we choose the time-frequency representation of the envelope and phase misfit of seismograms to assess the accuracy of the resulting seismograms since this provides the time evolution of the spectral content and allows for the clear separation of amplitude and phase errors obtained by the numerical method. Our results can be directly used to set up the necessary modelling parameters for practical applications, such as the minimum approximation order for a given mesh spacing to reach a desired accuracy. Finally, we apply our results to the well-acknowledged LOH.1 and LOH.3 problems of the SPICE Code Validation project, including heterogeneous material and the free surface boundary condition, and compare our solutions with those of other methods. In general, we want to stress the increasing importance of certain standard procedures to facilitate future code validations and comparisons of results in the community of numerical seismology. {\textcopyright} 2008 The Authors Journal compilation {\textcopyright} 2008 RAS.},
	keywords = {Computational seismology, Numerical approximations and analysis, Numerical solutions, Time series analysis, Wave propagation, Wavelet transform},
	issn = {0956540X},
	doi = {10.1111/j.1365-246X.2008.03781.x},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-43749087643\&partnerID=tZOtx3y1},
	author = {K{\"a}ser, Martin and Hermann, Verena and de la Puente, Josep}
}
@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}
}
@article {Dumbser2007,
	title = {{Arbitrary high-order finite volume schemes for seismic wave propagation on unstructured meshes in 2D and 3D}},
	journal = {Geophysical Journal International},
	volume = {171},
	number = {2},
	year = {2007},
	pages = {665{\textendash}694},
	abstract = {We 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. {\textcopyright} 2007 The Authors Journal compilation {\textcopyright} 2007 RAS.},
	keywords = {ADER approach, Attenuation, Finite volume schemes, High-order accuracy, Tetrahedral meshes, Viscoelasticity},
	issn = {0956540X},
	doi = {10.1111/j.1365-246X.2007.03421.x},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-35348851376\&partnerID=tZOtx3y1},
	author = {Dumbser, Michael and K{\"a}ser, Martin and de la Puente, Josep}
}
