  We analyse a new subdomain scheme for a time-spectral method for solving
initial boundary value problems. Whilst spectral methods are commonplace for
spatially dependent systems, finite difference schemes are typically applied
for the temporal domain. The Generalized Weighted Residual Method (GWRM) is a
fully spectral method in that it spectrally decomposes all specified domains,
including the temporal domain, with multivariate Chebyshev polynomials. The
Common Boundary-Condition method (CBC) is a spatial subdomain scheme that
solves the physical equations independently from the global connection of
subdomains. It is here evaluated against two finite difference methods. For the
linearised Burger equation the CBC-GWRM is $\sim30\%$ faster and $\sim50\%$
more memory efficient than the semi implicit Crank-Nicolson method at a maximum
error $\sim10^{-5}$. For a forced wave equation the CBC-GWRM manages to average
efficiently over the small time-scale in the entire temporal domain. The
CBC-GWRM is also applied to the linearised ideal magnetohydrodynamic (MHD)
equations for a screw pinch equilibrium. The growth rate of the most unstable
mode was efficiently computed with an error $<0.1\%$.
