With the advent of very large scale parallel computer, having millions of processing cores, using the time direction in evolution problems for parallelization has become an active field of research. Most of the methods developed for this purpose are iterative, like parareal algorithm or waveform relaxation methods based on domain decomposition.
We present here a mathematical analysis of a direct method to parallelize in time, proposed by Maday and Ronquist in 2007. It is based on the diagonalization of the time stepping matrix. We propose an optimization strategy for the choice of the time-steps, and show promising results for the heat equation in two dimensions.