论文标题
波动方程的伪谱时间域（PSTD）方法：实现离散正弦和余弦变换的边界条件
Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms
论文作者
论文摘要
伪谱时间域（PSTD）方法被广泛用于声学的许多分支中，用于波动的数值解，包括生物医学超声和地震学。特别是使用傅立叶搭配光谱法具有许多计算优势。但是，使用离散的傅立叶基础也固有地限于解决周期性边界条件的问题。在这里，描述了基于正弦或余弦基础的光谱搭配方法家族。这些保留了傅立叶搭配方法的计算优势，但允许施加同质的dirichlet（声音柔软）和诺伊曼（Sound-Hard）边界条件。使用离散的正弦和余弦变换对基本函数权重计算，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施光谱方法的实用详细信息。然后，通过在边界条件不同组合的矩形结构域中的波方程解决方程来说明该技术。还使用Dirichlet和Neumann边界条件的解决方案的加权求和也证明了具有任意实际反射系数或边界的边界扩展。
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.
