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Discontinuous Galerkin Methods for Wave Propagation in Poroelastic Media

De la Puente, Josep, Michael Dumbser, Martin Käser, and Heiner Igel (2008), Discontinuous Galerkin Methods for Wave Propagation in Poroelastic Media, Geophysics, 73(5), T77-T97, doi:10.1190/1.2965027.

We present a new numerical method to solve the heterogeneous poroelastic wave equation with arbitrary high order accuracy in space and time on unstructured tetrahedral meshes. By using Biot's equations and Darcy's Dynamic Laws we build a scheme able to successfully model wave propagation in fluid-saturated porous media where anisotropy of the pore structure is allowed. The Discontinuous Galerkin method, in contrast to classical Finite Element methods, allows for discontinuities of the piecewise polynomial approximation of the unknowns at element interfaces. This is then solved by making use of the well-established theory of Finite Volumes and numerical fluxes across element interfaces obtained by the solution of derivative Riemann problems. A continuous ADER time integration is used for the high-frequency and inviscid cases, while a space-time discontinuous scheme is applied for the low-frequency case. For both approaches 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. Numerical convergence studies confirm that the new schemes provide arbitrary high order accuracy even on unstructured tetrahedral meshes and shows that computational cost and storage space can be reduced by higher order schemes. The proposed method is able to solve accurately the low-frequency range for which the slow pressure waves show a diffusive behavior. For that case we compare the time-discontinuous approach to a fraction-step technique and show the the enhanced qualitative results of the first case. The validation of the method is achieved by reproducing known analytical solutions and comparing to results obtained by other existing methodologies.
  author = {Josep de la Puente and Michael Dumbser and Martin K{\"a}ser and Heiner Igel},
  journal = {Geophysics},
  number = {5},
  pages = {T77-T97},
  title = {{Discontinuous Galerkin Methods for Wave Propagation in Poroelastic Media}},
  volume = {73},
  year = {2008},
  doi = {10.1190/1.2965027},
%0 Journal Article
%A de la Puente, Josep
%A Dumbser, Michael
%A Käser, Martin
%A Igel, Heiner
%D 2008
%N 5
%V 73
%J Geophysics
%P T77-T97
%T Discontinuous Galerkin Methods for Wave Propagation in Poroelastic Media
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