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Pseudospectral time domain (PSTD) methods are widely used in many branches ofacoustics for the numerical solution of the wave equation, including biomedicalultrasound and seismology. The use of the Fourier collocation spectral methodin particular has many computational advantages. However, the use of a discreteFourier basis is also inherently restricted to solving problems with periodicboundary conditions. Here, a family of spectral collocation methods based onthe use of a sine or cosine basis is described. These retain the computationaladvantages of the Fourier collocation method but instead allow homogeneousDirichlet (sound-soft) and Neumann (sound-hard) boundary conditions to beimposed. The basis function weights are computed numerically using the discretesine and cosine transforms, which can be implemented using O(N log N)operations analogous to the fast Fourier transform. Practical details of how toimplement spectral methods using discrete sine and cosine transforms areprovided. The technique is then illustrated through the solution of the waveequation in a rectangular domain subject to different combinations of boundaryconditions. The extension to boundaries with arbitrary real reflectioncoefficients or boundaries that are non-reflecting is also demonstrated usingthe weighted summation of the solutions with Dirichlet and Neumann boundaryconditions.

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms