Pseudospectral time-domain method

In computational physics, pseudospectral time-domain method (PSTD) is a numerical analysis technique for simulating wave propagation. Being a part of the general class of pseudo-spectral methods, it is an extension of the finite-difference time-domain method (FDTD): in PSTD, the spatial derivative terms of the wave equation are evaluated in the spectral domain using orthogonal bases, such as Fourier or Chebyshev bases while the problem is temporally discretized. Introduced in the late 1990s, PSTD is widely used in acoustic, electromagnetic and geophysical simulations of complex and large-scale media, with a wide variety of applications such as photoacoustic imaging, geophysical imaging, light scattering, plasma physics and ultrasonics. The method is implemented in open-source acoustics codes such as k-Wave and openPSTD.

Theory The main concepts behind PSTD can be illustrated through the one-dimensional scalar wave equation:

∂ 2 u ( x , t ) ∂ x 2 + 1 v 2 ∂ 2 u ( x , t ) ∂ t 2 = 0 {\displaystyle {\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}+{\frac {1}{v^{2}}}{\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}=0}

where u ( x , t ) {\displaystyle u(x,t)} is the field component and v {\displaystyle v} is the wave speed. At a fixed point in time, this field component can be decomposed into an orthogonal basis, such as the Fourier basis. The main concept behind the PSTD algorithm is evaluating the spatial derivatives in this spectral-domain, while using finite difference approximations to compute temporal derivatives. As an example, the derivatives can be represented in the Fourier basis as:

∂ u ( x , t ) ∂ x = ∂ ∂ x ∑ n = − ∞ ∞ u ~ n ( t ) e i k n x = ∑ n = − ∞ ∞ i k n u ~ n ( t ) e i k n x = F − 1 [ i k F [ u ( x , t ) ] ] {\displaystyle {\frac {\partial u(x,t)}{\partial x}}={\frac {\partial }{\partial x}}\sum _{n=-\infty }^{\infty }{\tilde {u}}_{n}(t)e^{ik_{n}x}=\sum _{n=-\infty }^{\infty }ik_{n}{\tilde {u}}_{n}(t)e^{ik_{n}x}={\mathcal {F}}^{-1}\left[ik{\mathcal {F}}\left[u(x,t)\right]\right]}

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