Stable Parareal in Time Method for First- and Second-Order Hyperbolic Systems

The parareal in time algorithm allows one 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 one 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 fatally large lack of accuracy of the plain parareal in time algorithm, which does not conserve invariant quantities. A somewhat similar difficulty occurs for problems where the solution lacks regularity, either initially or during the evolution, as is the case for hyperbolic systems of conservation laws. In this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them. We use the new method to solve a linear wave equation and a nonlinear Burgers' equation. The results illustrate the stability of this variant of the parareal in time algorithm.

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