Abstract
Abstract
The computation of wave-energy distributions in the mid-to-high frequency regime can be reduced to ray-tracing calculations. Solving the ray-tracing problem in terms of an operator equation for the energy density leads to an inhomogeneous equation which involves a Perron–Frobenius operator defined on a suitable Sobolev space. Even for fairly simple geometries, let alone realistic scenarios such as typical boundary value problems in room acoustics or for mechanical vibrations, numerical approximations are necessary. Here we study the convergence of approximation schemes by rigorous methods. For circular billiards we prove that convergence of finite-rank approximations using a Fourier basis follows a power law where the power depends on the smoothness of the source distribution driving the system. The relevance of our studies for more general geometries is illustrated by numerical examples.
Funder
Engineering and Physical Sciences Research Council
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Cited by
2 articles.
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1. Ray-Tracing the Ulam Way;Integral Methods in Science and Engineering;2023
2. A Direction Preserving Discretization for Computing Phase-Space Densities;SIAM Journal on Scientific Computing;2021-01