Symmetric parareal algorithms for hamiltonian systems

Dai, Xiaoying; Le Bris, Claude; Legoll, Frédéric; Maday, Yvon

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 717-742.

The parareal in time algorithm allows for efficient parallel numerical simulations of time-dependent problems. It is based on a decomposition of the time interval into subintervals, and on a predictor-corrector strategy, where the propagations over each subinterval for the corrector stage are concurrently performed on the different processors that are available. In this article, we are concerned with the long time integration of Hamiltonian systems. Geometric, structure-preserving integrators are preferably employed for such systems because they show interesting numerical properties, in particular excellent preservation of the total energy of the system. Using a symmetrization procedure and/or a (possibly also symmetric) projection step, we introduce here several variants of the original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E 66 (2002) 057701; G. Bal and Y. Maday, A parareal time discretization for nonlinear PDE's with application to the pricing of an American put, in Recent developments in domain decomposition methods, Lect. Notes Comput. Sci. Eng. 23 (2002) 189-202; J.-L. Lions, Y. Maday and G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001) 661-668.] that are better adapted to the Hamiltonian context. These variants are compatible with the geometric structure of the exact dynamics, and are easy to implement. Numerical tests on several model systems illustrate the remarkable properties of the proposed parareal integrators over long integration times. Some formal elements of understanding are also provided.

Classification : 65L05, 65P10, 65Y05
Mots clés : parallel integrators, hamiltonian dynamics, long-time integration, symmetric algorithms, symmetric projection, geometric integration

[1] H.C. Andersen, Rattle: a velocity version of the Shake algorithm for molecular dynamics calculations. J. Comput. Phys. 52 (1983) 24-34. | Zbl

[2] L. Baffico, S. Bernard, Y. Maday, G. Turinici and G. Zérah, Parallel in time molecular dynamics simulations. Phys. Rev. E 66 (2002) 057701.

[3] G. Bal, On the convergence and the stability of the parareal algorithm to solve partial differential equations, in Domain decomposition methods in science and engineering, edited by R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund and J. Xu. Springer Verlag, Lect. Notes Comput. Sci. Eng. 40 (2005) 425-432. | MR | Zbl

[4] G. Bal and Y. Maday, A parareal time discretization for nonlinear PDE's with application to the pricing of an American put, in Recent developments in domain decomposition methods, edited by L.F. Pavarino and A. Toselli. Springer Verlag, Lect. Notes Comput. Sci. Eng. 23 (2002) 189-202. | MR | Zbl

[5] G. Bal and Q. Wu, Symplectic parareal, in Domain decomposition methods in science and engineering, edited by U. Langer, M. Discacciati, D.E. Keyes, O.B. Widlund and W. Zulehner. Springer Verlag, Lect. Notes Comput. Sci. Eng. 60 (2008) 401-408. | MR | Zbl

[6] A. Bellen and M. Zennaro, Parallel algorithms for initial value problems for nonlinear vector difference and differential equations. J. Comput. Appl. Math. 25 (1989) 341-350. | MR | Zbl

[7] G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74 (1994) 1117-1143. | MR | Zbl

[8] L.A. Berry, W. Elwasif, J.M. Reynolds-Barredo, D. Samaddar, R. Sanchez and D.E. Newman, Event-based parareal: A data-flow based implementation of parareal. J. Comput. Phys. 231 (2012) 5945-5954.

[9] K. Burrage, Parallel and sequential methods for ordinary differential equations, Numerical Mathematics and Scientific Computation, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1995). | MR | Zbl

[10] K. Burrage, Parallel methods for ODEs. Advances Comput. Math. 7 (1997) 1-3. | MR | Zbl

[11] P. Chartier and B. Philippe, A parallel shooting technique for solving dissipative ODE's. Computing 51(3-4) (1993) 209-236. | MR | Zbl

[12] X. Dai, C. Le Bris, F. Legoll and Y. Maday, Symmetric parareal algorithms for Hamiltonian systems, arXiv:preprint 1011.6222. | MR

[13] C. Farhat, J. Cortial, C. Dastillung and H. Bavestrello, Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses. Int. J. Numer. Meth. Engng. 67 (2006) 697-724. | MR | Zbl

[14] P. Fischer, F. Hecht and Y. Maday, A parareal in time semi-implicit approximation of the Navier Stokes equations, in Domain decomposition methods in science and engineering, edited by R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund and J. Xu. Springer Verlag Lect. Notes Comput. Sci. Eng. 40 (2005) 433-440. | MR

[15] D. Frenkel and B. Smit, Understanding molecular simulation, from algorithms to applications, 2nd ed., Academic Press (2002). | Zbl

[16] M. Gander and S. Vandewalle, On the superlinear and linear convergence of the parareal algorithm, in Proceedings of the 16th International Conference on Domain Decomposition Methods, January 2005, edited by O. Widlund and D. Keyes. Springer, Lect. Notes Comput. Sci. Eng. 55 (2006) 291-298. | MR | Zbl

[17] M. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29 (2007) 556-578. | MR | Zbl

[18] I. Garrido, B. Lee, G.E. Fladmark and M.S. Espedal, Convergent iterative schemes for time parallelization. Math. Comput. 75 (2006) 1403-1428. | MR | Zbl

[19] W. Hackbusch, Parabolic multigrid methods, Computing methods in applied sciences and engineering VI (Versailles, 1983), North-Holland, Amsterdam (1984) 189-197. | MR | Zbl

[20] E. Hairer, Symmetric projection methods for differential equations on manifolds. BIT 40 (2000) 726-734. | MR | Zbl

[21] E. Hairer and C. Lubich, The life span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462. | MR | Zbl

[22] E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer Ser. Comput. Math. 31 (2002). | MR | Zbl

[23] P. Joly, Numerical methods for elastic wave propagation, in Waves in nonlinear pre-stressed materials, edited by M. Destrade and G. Saccomandi. Springer-Verlag (2007) 181-281. | MR | Zbl

[24] P. Joly, The mathematical model for elastic wave propagation. Effective computational methods for wave propagation, in Numer. Insights, Chapman & Hall/CRC (2008) 247-266. | MR | Zbl

[25] J. Laskar, A numerical experiment on the chaotic behavior of the Solar system. Nature 338 (1989) 237-238.

[26] J. Laskar, Chaotic diffusion in the Solar system. Icarus 196 (2008) 1-15.

[27] B. Leimkuhler and S. Reich, Simulating Hamiltonian dynamics. Cambridge University Press (2004). | MR | Zbl

[28] B.J. Leimkuhler and R.D. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems. J. Comput. Phys. 112 (1994) 117-125. | MR | Zbl

[29] E. Lelarasmee, A.E. Ruehli and A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. CAD of IC Syst. 1 (1982) 131-145.

[30] J.-L. Lions, Y. Maday and G. Turinici, A parareal in time discretization of PDE's. C. R. Acad. Sci. Paris, Ser. I 332 (2001) 661-668. | MR | Zbl

[31] Y. Maday, The parareal in time algorithm, in Substructuring Techniques and Domain Decomposition Methods, edited by F. Magoulès. Chapt. 2, Saxe-Coburg Publications, Stirlingshire, UK (2010) 19-44. doi:10.4203/csets.24.2

[32] Y. Maday and 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. | MR | Zbl

[33] Y. Maday and G. Turinici, A parallel in time approach for quantum control: the parareal algorithm. Int. J. Quant. Chem. 93 (2003) 223-228.

[34] Y. Maday and G. Turinici, The parareal in time iterative solver: a further direction to parallel implementation, in Domain decomposition methods in science and engineering, edited by R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund and J. Xu. Springer Verlag, Lect. Notes Comput. Sci. Eng. 40 (2005) 441-448. | MR | Zbl

[35] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1999). | MR | Zbl

[36] S. Reich, Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36 (1999) 1549-1570. | MR | Zbl

[37] J.-P. Ryckaert, G. Ciccotti and H.J.C. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes. J. Comput. Phys. 23 (1977) 327-341.

[38] P. Saha, J. Stadel and S. Tremaine, A parallel integration method for Solar system dynamics. Astron. J. 114 (1997) 409-414.

[39] P. Saha and S. Tremaine, Symplectic integrators for solar system dynamics. Astron. J. 104 (1992) 1633-1640.

[40] J.M. Sanz-Serna and M.P. Calvo, Numer. Hamiltonian Problems. Chapman & Hall (1994). | MR | Zbl

[41] G.A. Staff and E.M. Rønquist, Stability of the parareal algorithm, in Domain decomposition methods in science and engineering, edited by R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund and J. Xu. Springer Verlag, Lect. Notes Comput. Sci. Eng. 40 (2005) 449-456. | MR | Zbl

[42] A. Toselli and O. Widlund, Domain decomposition methods-algorithms and theory. Springer Ser. Comput. Math. 34 (2005). | MR | Zbl