Developing Perfectly Matched Layer method to solve Heat Equation numerically

Abstract

Perfectly Matched Layer (PML) techniques, although having been studied extensively to study systems with nonrestrictive boundaries in many physical fields, are not readily adaptable to the study of thermodynamic systems governed by the heat equation. Using the explicit finite difference method, we can easily describe systems with perfectly absorbing or reflecting boundaries. These, however, are highly idealized physical states, so we have begun extending our abilities to simulating more realistic physical situations by defining arbitrary spatial differential operators, which govern how heat at the boundaries of the system of interest propagates. As a first attempt, we transformed the linear spatial domain to a trigonometric domain to combat reflection from boundaries, but this is quite crude. We then approached the problem using Fourier Transforms in order to define the problem in a heat distribution's frequency space, damp excessive heat past the boundaries, and transform back to position space. Initial results using the finite difference method verify the current computer simulation's ability to solve problems with ideal circumstances, and the development of a PML method which accurately simulates nonrestrictive boundaries and can be easily translated into a computer algorithm is in progress.