  Many multiscale problems have a high contrast, which is expressed as a very
large ratio between the media properties. The contrast is known to introduce
many challenges in the design of multiscale methods and domain decomposition
approaches. These issues to some extend are analyzed in the design of spatial
multiscale and domain decomposition approaches. However, some of these issues
remain open for time dependent problems as the contrast affects the time
scales, particularly, for explicit methods. For example, in parabolic
equations, the time step is $dt=H^2/\kappa_{max}$, where $\kappa_{max}$ is the
largest diffusivity. In this paper, we address this issue in the context of
parabolic equation by designing a splitting algorithm. The proposed splitting
algorithm treats dominant multiscale modes in the implicit fashion, while the
rest in the explicit fashion. The unconditional stability of these algorithms
require a special multiscale space design, which is the main purpose of the
paper. We show that with an appropriate choice of multiscale spaces we can
achieve an unconditional stability with respect to the contrast. This could
provide computational savings as the time step in explicit methods is adversely
affected by the contrast. We discuss some theoretical aspects of the proposed
algorithms. Numerical results are presented.
