  This article proposes spectral numerical methods to solve the time evolution
of convection problems with viscosity strongly depending on temperature at
infinite Prandtl number. Although we verify the proposed techniques just for
viscosities that depend exponentially on temperature, the methods are
extensible to other dependence laws. The set-up is a 2D domain with periodic
boundary conditions along the horizontal coordinate. This introduces a symmetry
in the problem, the O(2) symmetry, which is particularly well described by
spectral methods and motivates the use of these methods in this context. We
examine the scope of our techniques by exploring transitions from stationary
regimes towards time dependent regimes. At a given aspect ratio stable
stationary solutions become unstable through a Hopf bifurcation, after which
the time-dependent regime is solved by the spectral techniques proposed in this
article.
