Parallel solution in time of ODEs: some achievements and perspectives

The parallel solution of initial value problems for ordinary differential equations (ODE-IVPs) has received much interest from many researchers in the past years. In general, the possibility of using parallel computing in this setting concerns different aspects of the numerical solution of ODEs, depending on the parallel platform to be used and/or the complexity of the problem to be solved. In particular, in this paper we examine possible extensions of a parallel method previously proposed in the mid-nineties [P. Amodio, L. Brugnano, Parallel implementation of block boundary value methods for ODEs, J. Comput. Appl. Math. 78 (1997) 197–211; P. Amodio, L. Brugnano, Parallel ODE solvers based on block BVMs, Adv. Comput. Math. 7 (1997) 5–26], and analyze its connections with subsequent approaches to the parallel solution of ODE-IVPs, in particular the “Parareal” algorithm proposed in [J.L. Lions, Y. Maday, G. Turinici, Résolution d'EDP par un schéma en temps “pararéel”, C. R. Acad. Sci. Paris, Ser. I 332 (2001) 661–668; Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 387–392].