Summary: DISCRETE TRANSPARENT BOUNDARY CONDITIONS FOR
WIDE ANGLE PARABOLIC EQUATIONS: FAST CALCULATION
AND APPROXIMATION
M. Ehrhardt and A. Arnold
Matthias Ehrhardt, Technische Universit¨at Berlin, Institut f¨ur Mathematik, Str. des 17.Juni
136, D­10623 Berlin, Germany
e-mail: ehrhardt@math.tu-berlin.de
Anton Arnold, Institut f¨ur Numerische Mathematik, Universit¨at M¨unster, Einsteinstr. 62,
D­48149 M¨unster, Germany
e-mail: anton.arnold@math.uni-muenster.de
This paper is concerned with the efficient implementation of transparent boundary condi-
tions (TBCs) for wide angle parabolic equations (WAPEs) assuming cylindrical symmetry.
In [1] a discrete TBC of convolution type was derived from the fully discretized whole­space
problem that is reflection­free and yields an unconditionally stable scheme. Since the dis-
crete TBC includes a convolution with respect to range with a weakly decaying kernel, its
numerical evaluation becomes very costly for long-range simulations.
As a remedy we construct new approximative transparent boundary conditions involving
exponential sums as an approximation to the convolution kernel. This special approxima-
tion enables us to use a fast evaluation of the convolution type boundary condition.
This new approach was outlined in detail in [2] for the standard "parabolic" equation.

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien
Ehrhardt, Matthias - Institut für Mathematik, Technische Universität Berlin

Collections: Mathematics