Topic : Numerically solving infinite domain problems.
Meeting Time : Thursday, 1:15
Meeting Place : Applied Mathematics Library x(Guggenheim 408D)
A review of some of the issues related to numerically solving infinite
domain problems is
This week's meeting (February 26)
Abstract
We are interested in solving the Poisson equation for some open region
numerically using a domain decomposition method. We compare the Schwarz
iteration, the Dirichlet-Neumann iteration, and a method using nonlocal
operators similar to the Calderon-Seeley projection operators. We give a
method for computing optimal transmission operators given arbitrary domain
decompositions. We also have numerical results for certain simple cases on
the sphere.
I'll describe a type of exact far-field boundary condition which makes use of a Dirichlet-to-Neumann map imposed on a circular domain in the far-field. This method leads to a boundary condition which is exact for the first m Fourier modes of the solution in the far-field. I'll then discuss some imlementation details, and show how this method can be used for standard Poisson problems, as well as problems in from potential theory.
Discussion of the Helmholtz equation and the need for radiation boundary conditions.
Discussion of the Perfectly Matched Layers (PML)
Two papers which should be useful for discussion :
S. Abarbanel and D. Gottlieb, A Mathematical Analysis of the PML
Method, J. of Comput. Phys,
S. Abarbanel, D. Gottlieb and J. S. Hesthaven, Well-posed Perfectly Matched
Layers for Advective Acoustics,
J. of Comput. Phys,
You can reach JCP online via this link.
I will continue our discussion of absorbing boundary conditions and give an introduction to Perfectly Matched Layers (PML) for time-dependent acoustics problems. I'll approach this from the standpoint of wave-propagation algorithms (as in CLAWPACK), where I implemented the PML presented by Hu in the paper mentioned below and get considerably better results than with simple extrapolation boundary conditions.
Fang Q. Hu, A Stable, Perfectly Matched Layer for Linearized Euler Equations in Unsplit Physical Variables, JCP 173 (2001), 455-480.
S. Abarbanel, D. Gottlieb and J. S. Hesthaven, Well-posed Perfectly Matched Layers for Advective Acoustics, J. of Comput. Phys, 154 (1999), pp. 266-283, .
You can reach JCP online via this link.
Randy will continue to discuss PML for wave propagation algorithms.
See papers referenced above for details on what he plans to discuss.
F. Q. Hu has several papers available for downloading via this link.
Peter will touch on a few of the major issues on infinite domain boundary problems in atmospheric science, and then describe in some detail artificial boundary conditions which permit the radiation of vertically propagating internal gravity waves.