An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes IV: Anisotropy

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’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 modeling of triclinic anisotropy, the most general crystalline symmetry class. The proposed method combines the Discontinuous Galerkin method with the ADER time integration approach using arbitrary high order derivatives of the piecewise polynomial representation of the unknown solution. In contrast to classical Finite Element methods discontinuities of this piecewise polynomial approximation are allowed at element interfaces, which allows for 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 ADER time integration technique the scheme provides the same approximation order in space and time automatically. Furthermore, through the projection of the tetrahedral elements of the physical space onto a canonical reference tetrahedron an efficient implementation is possible as many three-dimensional integral computations can be carried out analytically beforehand. A numerical convergence study confirms that the new scheme provides arbitrary high order accuracy even on unstructured tetrahedral meshes and shows that computational cost and storage space can be reduced by higher order schemes. We additionally include a way to couple anisotropy with viscoelastic attenuation based on the Generalized Maxwell Body rheology and the mean and deviatoric stress concepts. Besides, we present a new Godunov-type numerical flux for anisotropic material and compare its accuracy with a computationally simpler Rusanov flux. Finally, we validate the new scheme by comparing the results of our simulations to an analytic solution as well as to Spectral Element computations.

@article{id604,
  author = {de la Puente, J. and K\"aser, M. and Dumbser, M. and Igel, H.},
  doi = {10.1111/j.1365-246X.2007.03381.x},
  journal = {Geophysical Journal International},
  language = {en},
  number = {3},
  pages = {1210-1228},
  title = {An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes IV: Anisotropy},
  url = {http://www.geophysik.uni-muenchen.de/{\textasciitilde}igel/PDF/delapuente\_gji\_2006.pdf},
  volume = {169},
  year = {2007},
}

%O Journal Article
%A de la Puente, J.
%A Käser, M.
%A Dumbser, M.
%A Igel, H.
%R 10.1111/j.1365-246X.2007.03381.x
%J Geophysical Journal International
%G en
%N 3
%P 1210-1228
%T An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes IV: Anisotropy
%U http://www.geophysik.uni-muenchen.de/~igel/PDF/delapuente_gji_2006.pdf
%V 169
%D 2007