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 O(Nlog N) 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.

中文翻译：

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波动方程的伪谱时域 (PSTD) 方法：用离散正弦和余弦变换实现边界条件

伪谱时域 (PSTD) 方法广泛用于声学的许多分支中，用于求解波动方程的数值解，包括生物医学超声和地震学。傅里叶搭配谱法的使用尤其具有许多计算优势。然而，离散傅里叶基的使用本质上也仅限于解决具有周期性边界条件的问题。这里，描述了基于使用正弦或余弦基的一系列光谱搭配方法。这些保留了傅里叶搭配方法的计算优势，但允许施加齐次 Dirichlet（声音软）和 Neumann（声音硬）边界条件。使用离散正弦和余弦变换以数值方式计算基函数权重，可以使用以下方法实现○(ñ日志 ñ)类似于快速傅里叶变换的操作。提供了如何使用离散正弦和余弦变换实现频谱方法的实际细节。然后通过求解受边界条件不同组合的矩形域中的波动方程来说明该技术。使用具有 Dirichlet 和 Neumann 边界条件的解的加权求和也证明了对具有任意实反射系数或非反射边界的边界的扩展。