Publication Details
Pseudospectral Time-Domain (PSTD) Methods for the Wave Equation: Realizing Boundary Conditions with Discrete Sine and Cosine Transforms
Boundary conditions; discrete cosine transform; discrete sine transform;
pseudospectral.
Pseudospectral time domain (PSTD) methods are widely used in many branches of
acoustics for the numerical solution of the wave equation, including biomedical
ultrasound and seismology. The use of the Fourier collocation spectral method in
particular has many computational advantages, including a reduced number of grid
points required for accurate simulations. However, the use of a discrete Fourier
basis is also inherently restricted to solving problems with periodic
boundary conditions. This means that waves exiting one side of the domain
reappear on the opposite side. Practically, this is usually overcome by
implementing a perfectly matched layer to simulate free-field conditions.
However, in some cases, other boundary conditions are required, and these are
not straightforward to implement. Here, a family of spectral collocation methods
based on the use of a sine or cosine basis is described. These retain the
computational advantages of the Fourier collocation method but instead allow
homogeneous Dirichlet (sound-soft) and Neumann (sound-hard) boundary conditions
to be imposed. The basis function weights are computed numerically using the
discrete sine and cosine transforms, which can be implemented using O(N log N )
operations analogous to the fast Fourier transform. The different combinations of
discrete symmetry give rise to sixteen possible discrete trigonometric
transforms. The properties of these transforms are described, and practical
details of how to implement spectral methods using a sine and cosine basis are
provided. The technique is then illustrated through the solution of the wave
equation in a rectangular domain subject to different combinations of boundary
conditions. The extension to boundaries with arbitrary reflection coefficients or
boundaries that are non-reflecting is also demonstrated using the
weighted summation of the solutions with Dirichlet and Neumann boundary
conditions.
@article{BUT168531,
author="Elliott {Wise} and Jiří {Jaroš} and Ben {Cox} and Bradley {Treeby}",
title="Pseudospectral Time-Domain (PSTD) Methods for the Wave Equation: Realizing Boundary Conditions with Discrete Sine and Cosine Transforms",
journal="Journal of Theoretical and Computational Acoustics",
year="2021",
volume="29",
number="4",
pages="2050021--2050021",
doi="10.1142/S2591728520500218",
issn="2591-7285",
url="https://arxiv.org/abs/2005.00322"
}