Seismic Waves in Heterogeneous Material: Sub-cell Resolution of the Discontinuous Galerkin Method

We present an important extension of the arbitrary high-order Discontinuous Galerkin (DG) Finite-Element method to model two-dimensional elastic wave propagation in highly heterogeneous material. In this new approach we include space-variable coefficients to describe smooth or discontinuous material variations inside each element using the same numerical approximation strategy as for the velocity-stress variables in the formulation of the elastic wave equation. The combination of the DG method with a time integration scheme based on the solution of Arbitrary accuracy DErivatives Riemann problems (ADER) still provides an explicit, one-step scheme which achieves arbitrary high-order accuracy in space and time. Compared to previous formulations the new scheme contains two additional terms in the form of volume integrals. We show that the increasing computational cost per element can be overcompensated due to the improved material representation inside each element as coarser meshes can be used which reduces the total number of elements and therefore computational time to reach a desired error level. We confirm the accuracy of the proposed scheme performing convergence tests and several numerical experiments considering smooth and highly heterogeneous material. As the approximation of the velocity and stress variables in the wave equation and of the material properties in the model can be chosen independently, we investigate the influence of the polynomial material representation on the accuracy of the synthetic seismograms with respect to computational cost. Moreover, we study the behaviour of the new method on strong material discontinuities, in the case where the mesh is not aligned with such a material interface. In this case second-order linear material approximation seems to be the best choice, with higher order intra-cell approximation leading to potential instable behaviour. For all test cases we validate our solution against the well-established standard 4th-order Finite Difference (FD) and Spectral Element Method (SEM).

@article{id1426,
  author = {Castro, Crist\'obal E. and K\"aser, Martin and Brietzke, Gilbert B.},
  doi = {10.1111/j.1365-246X.2010.04605.x},
  journal = {Geophys. J. Int.},
  language = {en},
  number = {1},
  pages = {250-264},
  title = {Seismic Waves in Heterogeneous Material:  Sub-cell Resolution of the Discontinuous Galerkin Method},
  volume = {182},
  year = {2010},
}

%O Journal Article
%A Castro, Cristóbal E.
%A Käser, Martin
%A Brietzke, Gilbert B.
%R 10.1111/j.1365-246X.2010.04605.x
%J Geophys. J. Int.
%G en
%N 1
%P 250-264
%T Seismic Waves in Heterogeneous Material:  Sub-cell Resolution of the Discontinuous Galerkin Method
%V 182
%D 2010