  A class of nonstandard pseudospectral time domain (PSTD) schemes for solving
time-dependent hyperbolic and parabolic partial differential equations (PDEs)
is introduced. These schemes use the Fourier collocation spectral method to
compute spatial gradients and a nonstandard finite difference scheme to
integrate forwards in time. The modified denominator function that makes the
finite difference time scheme exact is transformed into the spatial frequency
domain or k-space using the dispersion relation for the governing PDE. This
allows the correction factor to be applied in the spatial frequency domain as
part of the spatial gradient calculation. The derived schemes can be formulated
to be unconditionally stable, and apply to PDEs in any space dimension.
Examples of the resulting nonstandard PSTD schemes for several PDEs are given,
including the wave equation, diffusion equation, and convection-diffusion
equation.
