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

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 κ = 50 and κ = 100 , where κ is a normalized wave number, defined through κ = length of wave number vector × 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.