Address:Analysis und Scientific Computing
Vienna University of Technology
Wiedner Hauptstrasse 8-10
A-1040 Wien, Austria
Telephone: +43 - 1 - 58801 - 10176
Fax: +43 - 1 - 58801 - 11599
Email: bierwirt@aurora.anum.tuwien.ac.at
Ph.D. Adviser: Anton Arnold
Research: Numerical discretization for Schroedinger-type one-way wave equation
If the potential or refraction index in the 2D Helmholtz Equation (HE) depends only on one variable $z$ (i.e. one considers a layered media), it is possible to (exactly) factorize the HE and gain a Pseudodifferential-Equation for the outward moving waves. This evolution-equation is the so-called Schroedinger-type one-way wave equation (OWWE).
The Pseudodifferential-Operator that appears in the OWWE is
which is a
non-local operator!"Classical" algorithms that are based on this approach use a Taylor- or Pade-approximation of the square root to replace the Helmholtz square root operator (HSRO)
defined above by a "rational" operator. The advantages of these algorithms are stability and the local behavior of the numerical schemes. But it is not possible to show consistency of these schemes with respect to the OWWE, i.e. the computed solutions do not converge to a solution of the OWWE or HE.
Our aim is to skip the rough approximation of the HSRO and directly derive a difference scheme from the OWWE in order to get consistency. Stability of the numerical scheme will be achieved, if one uses the Crank-Nicolson scheme for the evolution direction under the assumption that the refraction index is real-valued. The current research is focused on the following topics:
Master thesis: The $\bar\partial$-Poincare Lemma on foliated spaces
Activities: