New parallel methods, based on the Schwarz and Schur domain decomposition technique, are proposed for time domain decomposition of Cauchy problem. Firstly, the initial value problem is transformed in a time boundary values problem. Then the demonstration of the pure linear divergence/convergence of the Schwarz algorithm, applied to systems of linear ODE, allows us to accelerate the convergence of the solution at the time slices boundaries with the Aitken's acceleration technique of the convergence. Secondly, a Schur complement method is developed for linear problem. Numerical results show the efficiency of the proposed method on the linear system of ODEs modeling the harmonic oscillator problem.
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