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Application of perfectly matched layer transformation to second order derivative

  1. Jul 25, 2010 #1
    Hello

    I am going to post here a problem that I've been working on for quite some time, and I haven't managed to obtain a good answer.

    To approximate free-field conditions in the numerical solution of wave equation PDEs, the following coordinate transformation is often applied to implement a Perfectly Matched Layer (PML):

    [tex]
    \[
    \frac{\partial }{{\partial x}} \to \frac{1}{{1 + \frac{{i\sigma (x)}}{\omega }}}\frac{\partial }{{\partial x}}
    \]
    [/tex]

    There is further information given in this application note:

    http://www-math.mit.edu/~stevenj/18.369/pml.pdf" [Broken]

    Normally this coordinate-stretching is applied to [tex]\partial /\partial x[/tex]

    But is it possible to apply the coordinate-stretching in the following fashion?

    [tex]
    \[
    \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{u}\frac{{\partial ^2 }}{{\partial x^2 }}
    \]
    [/tex]

    That is, the coordinate stretching is applied without recourse to this:

    [tex]
    \[
    \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{s}\frac{\partial }{{\partial x}}\left( {\frac{1}{s}\frac{\partial }{{\partial x}}} \right)
    \]
    [/tex]

    where

    [tex]
    \[
    s = 1 + \frac{{i\sigma (x)}}{\omega }
    \]
    [/tex]
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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