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An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes V: Local Time Stepping and p-Adaptivity

Dumbser, Michael, Martin Käser, and Eleuterio Toro (2007), An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes V: Local Time Stepping and p-Adaptivity, Geophysical Journal International, 171(2), 695-717, doi:10.1111/j.1365-246X.2007.03427.x.

Abstract
This article describes the extension of the ADER-DG method to treat locally
varying polynomial degress of the basis functions, so-called p-adaptivity,
as well as locally varying time steps that may be different from one element
to another.
The p-adaptive version of the scheme is useful in complex three-dimensional
models with small-scale features which have to be meshed with reasonably
small elements to capture the necessary geometrical details of interest. Using a
constant high polynomial degree of the basis functions in the whole computational
domain can lead to an unreasonably high CPU effort since good spatial resolution
at the surface may be already obtained by the fine mesh. Therefore, it can be more
adequate in some cases to use a lower order method in the small elements to reduce
the CPU effort without loosing much accuracy.
To further increase computational efficiency, we present a new local time stepping
(LTS) algorithm. For usual explicit time stepping schemes the element with the
smallest time step resulting from the stability criterion of the method will dictate
its time step to all the other elements of the computational domain. In contrast, by using local
time stepping, each element can use its optimal time step given by the local stability
condition.
Our proposed LTS algorithm for ADER-DG is very general and does not need any
temporal synchronization between the elements. Due to the ADER approach, accurate
time interpolation is automatically provided at the element interfaces such that the
computational overhead is very small and such that the method maintains the uniform high
order of accuracy in space and time as in the usual ADER-DG schemes with a globally constant
time step. However, the LTS ADER-DG method is computationally much more efficient for
problems with strongly varying element size or material parameters since it allows to reduce
the total number of element updates considerably. This holds especially for unstructured
tetrahedral meshes that contain strongly degenerate elements, so-called slivers.
We show numerical convergence results and CPU times for LTS ADER-DG schemes up to sixth
order in space and time on irregular tetrahedral meshes containing elements of very
different size and also on tetrahedral meshes containing slivers.
Further validation of the algorithm is provided by results obtained for the layer
over halfspace (LOH.1) benchmark problem proposed by the Pacific Earthquake Engineering
Research Center.
Finally, we present a realistic application on earthquake modelling and
ground motion prediction for the alpine valley of Grenoble.
BibTeX
@article{id891,
  author = {Michael Dumbser and Martin K{\"a}ser and Eleuterio Toro},
  journal = {Geophysical Journal International},
  number = {2},
  pages = {695-717},
  title = {{An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes V: Local Time Stepping and p-Adaptivity}},
  volume = {171},
  year = {2007},
  doi = {10.1111/j.1365-246X.2007.03427.x},
}
EndNote
%0 Journal Article
%A Dumbser, Michael 
%A Käser, Martin
%A Toro, Eleuterio
%D 2007
%N 2
%V 171
%J Geophysical Journal International
%P 695-717
%T An Arbitrary High Order Discontinuous Galerkin Method for Elastic Waves on Unstructured Meshes V: Local Time Stepping and p-Adaptivity
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Printed 08. Dec 2019 07:26