  This paper presents a new algorithm for the parallel in time (PiT) numerical
simulation of time dependent partial/ordinary differential equations. We
propose a reliable alternative to the well know parareal in time algorithm, by
formulating the parallel in time problem algebraically and solve it using an
adapted Bi-Conjugate gradient stabilized method. The proposed Parallel in time
Stable Bi-Conjugate algorithm (PiTSBiCG) has a great potential in stabilizing
the parallel resolution for a variety of problems. In this work, we describe
the mathematical approach to the new algorithm and provide numerical evidences
that show its superiority to the standard parareal method.
