1. 1 DICA, Università di Trento; 2 Institut f. Geophysik, ETH Zürich
The Discontinuous Galerkin Method for 3-D Elastic Wave Propagation and Kinematic Sources
SPICE
M. Käser1, M. Dumbser1, P.M. Mai2
1 DICA, Università di Trento; 2 Institut f. Geophysik, ETH Zürich
The DG-Method
The Semi-Discrete Scheme
Numerical Example
Considering the 3-D elastic wave equation in velocity-stress formulation leads to the hyper-bolic system of the form
where the vector of unknowns is given by
The numerical solution Qh of this system can
be approximated inside a tetrahedron by the linear combination of time-independent poly-nomial basis functions and time-dependent degrees of freedom, i.e.
For a 3rd order scheme, the 10 basis functions
are given as:
Multiplying the governing equation by a test function and integrating over a tetrahedron
gives
Transformation from a physical tetrahedron
into the reference tetrahedron improves the
computational efficiency as integrations can
be pre-computed analytically, such as:
The resulting scheme is then given through:
The time integration can be carried out by using a classical Runge-Kutta method or the new ADER
approach automatically leading to the same order of approximation in space and time.
Code validation through
wave excitation by finite
source rupture models
using a layered medium
and a set of well distri-
buted stations and their
reference seismograms.
Point Sources
Source terms acting only in a point can be described by the Dirac Delta distribution, where ,
and a source time function:
The space-time integral of the source term gives
where the test function Fk can be evaluated at any point location inside a tetrahedron of the compu-tational domain.
Mesh parameters:
Average edge length on surface: m
Average edge length on base: m
Total number of tetrahedrons:
Synthetic Seismograms
A comparison of the reference seismograms and
the DG synthetics. Higher order provides drastic
improvement.
Finite Source Rupture Models
The point source assumption can be applied to every
subfault of a finite source rupture model z z y h x x
3-D visualization of the elastic wave field
showing the vertical velocity component w
4 sec
8 sec
12 sec
Contact: Martin Käser, DICA, Università di Trento, Via Mesiano 77, I Trento,