- #1
nkinar
- 76
- 0
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"
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]
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"
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]
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