scholarly journals Representing arbitrary acoustic source and sensor distributions in Fourier collocation methods

2019 ◽  
Vol 146 (1) ◽  
pp. 278-288 ◽  
Author(s):  
Elliott S. Wise ◽  
B. T. Cox ◽  
Jiri Jaros ◽  
Bradley E. Treeby
Keyword(s):  
Collocation Methods ◽  
Acoustic Source ◽  
Fourier Collocation

scholarly journals Pseudospectral Time-Domain (PSTD) Methods for the Wave Equation: Realizing Boundary Conditions with Discrete Sine and Cosine Transforms

2020 ◽  
pp. 2050021
Author(s):  
Elliott S. Wise ◽  
Jiri Jaros ◽  
Ben T. Cox ◽  
Bradley E. Treeby
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Time Domain ◽  
Neumann Boundary ◽  
Rectangular Domain ◽  
Collocation Methods ◽  
Reflection Coefficients ◽  
Collocation Spectral Method ◽  
Spectral Collocation Methods ◽  
Fourier Collocation

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. However, the use of a discrete Fourier basis is also inherently restricted to solving problems with periodic boundary conditions. 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 [Formula: see text] operations analogous to the fast Fourier transform. Practical details of how to implement spectral methods using discrete sine and cosine transforms 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 real reflection coefficients or boundaries that are nonreflecting is also demonstrated using the weighted summation of the solutions with Dirichlet and Neumann boundary conditions.

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