🚀 go-pugleaf

In article <renns.703315913@sfu.ca> renns@fraser.sfu.ca (R Enns) writes:
>
>Does anyone have any experience with absorbing boundary conditions for PDE's?
>I am solving the generalized nonlinear Schroedinger equation: 
>
>         2i E    +   E    +    E    +    E     +  f(I=|E|^2) E  =  0
>             t        xx        yy        zz
>
>using the split-step Fourier method (SSFM) on a 128^3 mesh.  The interesting
>physics is occurring at the center of my lattice and I would like to absorb
>as much as possible of whatever hits the boundaries.  My pedestrian experiments
>with various boundary dampings simply create near total reflections.
>
>Any advice is sincerely appreciated.  Is 5-10% reflection achievable without
>much effort?
> 
>
>-- 
>
>Darran Edmundson
>darran@chaos.phys.sfu.ca

There are a couple of approaches that have been used in the chemistry
literature in the recent past (Journal of Chemical Physics).  I don't
have the specific references here, but can point out the years, anyway.
Back in ca. 1984, Kosloff and Cerjan used an exponentially decreasing
attenuator (one can use any smooth window function) placed on the last
several points of the relevant edge of the grid.  Empirically, they
found that 14-20 grid points were needed in order to avoid reflections.

Ca. 1988, Alvarellos and Metiu developed a variant that gracefully
bleeds off the wave function leaving the grid, propagating this portion
according to potential-less equations beyond the edge (which could be
done analytically in their initial application).

Somewhere in the time since then (1989-1990), Neuhauser and colleagues
have used a complex optical potential placed outside of the region of
the interesting physics in order to absorb the wave function.  I
believe this additional potential varies linearly with distance, but
I heard W. H. Miller contend at a conference a couple weeks ago that
there were some potential advantages (sorry) to nonlinear variation
with distance (not sure if this is published outside of the conference
proceedings).  One of my neighbors (D. J. Kouri at University of
Houston) is actively exploring this business also.

Bruce Johnson