  The parareal in time algorithm allows to perform parallel simulations of time
dependent problems. This algorithm has been implemented on many types of time
dependent problems with some success. Recent contributions have allowed to
extend the domain of application of the parareal in time algorithm so as to
handle long time simulations of Hamiltonian systems. This improvement has
managed to avoid the fatal large lack of accuracy of the plain parareal in time
algorithm consequence of the fact that the plain parareal in time algorithm
does not conserve invariants. A somehow similar difficulty occurs for problems
where the solution lacks regularity, either initially or in the evolution, like
for the solution to hyperbolic system of conservation laws. In this paper we
identify the problem of lack of stability of the parareal in time algorithm and
propose a simple way to cure it. The new method is used to solve a linear wave
equation and a non linear Burger's equation, the results illustrate the
stability of this variant of the parareal in time algorithm.
