Exponential decay of errors of a fundamental solution method applied to a reduced wave problem in the exterior region of a disc

Authors:
Fumihiro Chiba;Teruo Ushijima
Affiliations:
Department of Computer Science, Faculty of Electro-Communications, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan;The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

This paper concerns a fundamental solution method (FSM, in abbreviation) applied to a reduced wave problem in the exterior region of a disc. The convergent rate of approximate solutions to the exact one is proven to be asymptotically exponentially decreasing with respect to the number N of collocation points employed in an approximate problem. Using obtained FSM solutions we add two numerical tests: numerical estimate of errors including cases of high wave numbers; and visualization of total waves appeared in the scattering phenomena around a circular obstacle in the cases of @k=50 and @k=100, where @k is a normalized wave number, defined through @k= length of wave number vector x radius of the disc. In the second test, the total waves almost vanish behind the disc, seemingly corresponding to the phenomenon of shadowing in the classical literature of physics.