We developed a parallel time domain decomposition method to solve systems of ODEs based on the Aitken-Schwarz domain decomposition method only in time (Linel P, Tromeur-Dervout D. Aitken-Schwarz and schur complement methods for time domain decomposition. In: Parallel computing: from multicores and GPU's to Petascale. Advances in parallel computing, vol. 19; 2010. p. 75-82). The method transforms the initial time value problem into a time boundary values problem. This paper details the proof of the pure linear convergence of this DDM in case of linear system of ODEs and provides an optimized MPI parallel implementation when a regular size splitting of the time interval is performed. Then an extension of the method to the linear heat partial derivative equation is provided with a special care on boundary conditions to impose. A Fourier convergence analysis of this Schwarz DDM in time for this problem allows to limit the acceleration by Aitken's technique to Fourier low modes only.