Ierley et al. [13] solved a a class of nonlinear parabolic partial differential
equations with periodic boundary conditions using a Fourier representation in
space and a Chebyshev representation in time. Similarly as for the GWRM,
the Burger equation and other problems were solved with high resolution.
Tang and Ma [14] also used a spatial Fourier representation for solution of
parabolic equations, but introduced Legendre Petrov-Galerkin methods for
the temporal domain.
In 1994, Bar-Yoseph et al [15, 16] used space-time spectral element methods
for solving one-dimensional nonlinear advection-diffusion problems and
second order hyperbolic equations. Chebyshev polynomials were later employed
in space-time least-squares spectral element methods [17].
A theoretical analysis of Chebyshev solution expansion in time and onedimensional
space, for equal spectral orders, was given in [18]. The minimized
residuals employed were however different from those of the GWRM.
More recently Dehghan and Taleei [19] found solutions to the non-linear
Schr¨odinger equation, using a time-space pseudo-spectral method where the
basis functions in time and space were constructed as a set of Lagrange
interpolants.
Time-spectral methods feature high order accuracy in time. For implicit
finite difference methods, deferred correction may provide high order temporal
accuracy [20, 21]. A relatively recent approach to increase the temporal
efficiency of finite difference methods is time-parallelization via the parareal
algorithm [22]. This method, however, features rather low parallel efficiency
and improvements have been suggested, for example the use of spectral deferred
corrections [23].
An interesting Jacobian-free Newton-Krylov method for implicit timespectral
solution of the compressible Navier-Stokes equations has recently
been put forth by Attar [24].
A time-spectral method for periodic unsteady computations, using a
Fourier representation in time, was suggested in [25] and further developed
in [26] and [27]. A generalization to quasi-periodic problems was developed
in [28].
In summary, although time-spectral methods have been explored in various
forms by several authors during the last few decades, and were found
to be highly accurate, the GWRM as described in [3] has not been pursued.
The present work contributes to the evaluation of this method.
The structure of the paper is as follows. In section 2 we briefly review the
general GWRM formalism for solving a set of pde’s but subsequently restrict