Waves Impinging on a Planar Boundary

For a rectangular domain, with a point source excited by a Gaussian pulse and periodic boundary conditions in one direction, we compared (in the other direction) exact nonreflecting boundary conditions, as implemented by our method, with Berenger-type perfectly matched layers (PML).

On one nonreflecting boundary, as a function of time, the solution is illustrated below.

Waves Impinging on Planar Boundary
The global relative error of the computed solution, with our implementation of the exact nonreflecting boundary conditions and with exact Dirichlet boundary conditions, is shown below.
Plot showing errors over time of solution procedure with nonreflecting and Dirichlet boundary conditions.
Finally, the global relative error of the computed solution, with different perfectly matched layers, is shown. Of several layer thicknesses and attenuations that we tried, the three shown produced the lowest error.
Plot showing errors over time of solution procedure with Berenger-type PML.
Note that the error is quite substantial, approximately one hundredfold worse than with our implementation of the exact nonreflecting boundary conditions. In fact, while for the exact boundary conditions our implementation is fourth-order convergent, with PML the solution does not converge to the correct answer.

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Bradley K Alpert
Last modified: Fri Jul 14 12:11:40 MDT 2000