================================== Simple phase space filter examples

Introduction

In 2008, I (together with Avy Soffer) wrote a paper on Phase Space Filters, A stable absorbing boundary layer for anisotropic waves_.

.. _version: .. _A stable absorbing boundary layer for anisotropic waves: http://arxiv.org/abs/0805.2929

This git repo stores the source code for that paper.

Hint for possible TDPSF users

I have developed two separate versions of the TDPSF. One version, described in the paper Open Boundaries for the Nonlinear Schrodinger Equation_.

.. _Open Boundaries for the Nonlinear Schrodinger Equation: http://arxiv.org/abs/math/0609183

is based on the Windowed Fourier Transform. This version has provable error bounds, which is why it was the first paper I published, and why my Ph.D. thesis was based on it.

The other version_ (the one this repo is based on) uses a simpler and faster phase space filter. The accuracy of this filter is not proven, however in practice I have found it to be just as effective as the WFT based filter.

Additionally, the running time of the program is vastly superior, by a factor of 8-32x.

I strongly recommend that anyone who wishes to use phase space filters should use the simplified version.

Guide

Dependencies

You need python (probably 2.6 or later, but NOT 3.0), numpy, matplotlib and scipy.

http://python.org/

http://numpy.scipy.org/

http://www.scipy.org/

http://matplotlib.sourceforge.net/

schrodinger_test.py

The file schrodinger_test.py is the program I used to generate Figure 1 of Section 3.1. It implements the TDPSF for the one-dimensional Schrodinger equation.

This example is 1-dimensional.

Euler Solvers

The Euler equations are solved using a standard FFT-based spectral propagator. This example is 2-dimensional.

The file euler_exact.py solves the problem on a grid of size npoints * dx = 2048 * 0.125 = 256, which is large enough that the waves will not reach the boundary before t=50.

The file euler_subsonic.py solves the problem with the phase space filters.

The file error_check_euler.py measures the error by comparing the TDPSF simulations to the exact simulations. The simulations are stored in the relative directory euler_subsonic_$K and euler_subsonic_exact_$K, where $K is the frequency of the initial condition.

The file euler_subsonic_long.py solves the problem for a long time to measure the stability.

Maxwell Solvers

The naming for the maxwell examples is the same as for the euler examples.