Gaussian Beam Methods for the Helmholtz Equation
In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of non-trapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number $k$, both for sin...
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Zusammenfassung: | In this work we construct Gaussian beam approximations to solutions of the
high frequency Helmholtz equation with a localized source. Under the assumption
of non-trapping rays we show error estimates between the exact outgoing
solution and Gaussian beams in terms of the wave number $k$, both for single
beams and superposition of beams. The main result is that the relative local
$L^2$ error in the beam approximations decay as {$k^{-N/2}$ independent of
dimension and presence of caustics, for $N$-th order beams. |
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DOI: | 10.48550/arxiv.1304.1291 |