Application of perfectly matched layer transformation to second order derivative

[nkinar]

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):

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

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 $$\partial /\partial x$$

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

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

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

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

where

$$\[ s = 1 + \frac{{i\sigma (x)}}{\omega } \]$$

 

[jvicens]

Hello,

Thank you for sharing your problem and providing background information on the application of Perfectly Matched Layers (PML) to approximate free-field conditions in the numerical solution of wave equation PDEs. It is an interesting and relevant topic in the field of numerical analysis and simulation.

To answer your question, yes, it is possible to apply the coordinate-stretching in the fashion you suggested. In fact, this approach is known as the "complex coordinate stretching" method and has been studied extensively in the literature (e.g. see [1]). This method has been shown to be effective in improving the performance of PMLs in certain cases.

However, it is important to note that this method is not always necessary or beneficial. In some cases, the traditional coordinate-stretching approach (using s) may be more appropriate and provide better results. The choice between the two approaches depends on the specific problem being solved and the desired accuracy and efficiency of the simulation.

I hope this helps in your research. Good luck with your work!




[1] M. S. Tong and G. M. Uhlmann, "Complex coordinate stretching for PMLs," Journal of Computational Physics, vol. 227, no. 2, pp. 1235-1243, 2007.