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
Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms
摘要：
时域伪谱方法（PSTD）被广泛应用于声学的许多分支中，用于波动方程的数值求解，包括生物医学超声和地震学。傅里叶配置谱方法的使用尤其具有许多计算上的优点。然而，离散傅立叶基的使用也固有地限于解决具有周期性边界条件的问题。在这里，一个家庭的频谱配置方法的基础上使用的正弦或余弦的基础上进行说明。这些保留了傅立叶配置法的计算优势，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来数值计算基函数权重，这可以使用类似于快速傅立叶变换的O（N log N）操作来实现。提供了如何使用离散正弦和余弦变换实现谱方法的实际细节。该技术，然后说明通过在矩形域的波动方程的解决方案受到不同的边界条件的组合。扩展到任意的实际反射系数的边界或边界是非反射的也证明了使用加权求和的解决方案与Dirichlet和Neumann边界条件。
英文摘要：
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(N log 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 non-reflecting is also demonstrated using the weighted summation of the solutions with Dirichlet and Neumann boundary conditions.
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