I'm solving linearized Navier-Stokes equations with Perfectly Matched Layers in two spatial directions $x$ and $y$, but in the time-harmonic frequency $\omega$-domain, which is meant to be less restrictive than the time $t$-domain regarding PML parameters according to that reference. The complex coordinate stretching is typically

$$\frac{\partial}{\partial{x}} := \frac{1}{1 + i\sigma(x)/\omega}\frac{\partial}{\partial{x}}$$

Although my case is with fluid mechanics, I guess this question is generic and pertains to any kind of physics.

My PMLs in the streamwise direction seem to work as they should. However I cannot match the solution obtained in the '1D-x-PML' when a top (wall-normal $y$) PML is added.

Most of the energy to absorb is in the streamwise $x$ direction, where the boundary layer is developing. There is still some perturbation fields that I wish to vanish in the $y$ direction to prevent spurious reflections, but they do not have a 'wave-pattern' like in $x$.

The only rules of thumb I'm aware of is to use a PML width of at least half the wavelength of the wave you want to absorb and to have a sufficient number of points per wavelength. Is there anything else that people should know for multidimensional PMLs? Is it an issue to try and absorb a 'non wave-like' perturbation in one direction because you would need an 'infinite' PML since the wavelength is 'infinite'? Should the target reflection coefficient be heavily adjusted?


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