Symmetric parareal algorithms for Hamiltonian systems

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, Serie 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.

[1]  Yvon Maday,et al.  A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations , 2005 .

[2]  Charbel Farhat,et al.  Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses , 2006 .

[3]  Patrick Joly,et al.  Numerical Methods for Elastic Wave Propagation , 2007 .

[4]  E. Hairer,et al.  Simulating Hamiltonian dynamics , 2006, Math. Comput..

[5]  H. C. Andersen.  Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations , 1983 .

[6]  Joachim Stadel,et al.  A Parallel Integration Method for Solar System Dynamics , 1997 .

[7]  Yvon Maday,et al.  Parallel in time algorithms for quantum control: Parareal time discretization scheme , 2003 .

[8]  Y. Maday,et al.  A parareal in time procedure for the control of partial differential equations , 2002 .

[9]  Yvon Maday,et al.  A parallel in time approach for quantum control: the parareal algorithm , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[10]  S. Reich.  Backward Error Analysis for Numerical Integrators , 1999 .

[11]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[12]  Yvon Maday,et al.  The Parareal in Time Iterative Solver: a Further Direction to Parallel Implementation , 2005 .

[13]  Raúl Sánchez,et al.  Event-based parareal: A data-flow based implementation of parareal , 2012, J. Comput. Phys..

[14]  S. Tremaine,et al.  Symplectic integrators for solar system dynamics , 1992 .

[15]  Alberto L. Sangiovanni-Vincentelli,et al.  The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits , 1982, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[16]  Berend Smit,et al.  Understanding Molecular Simulation , 2001 .

[17]  J. Laskar.  A numerical experiment on the chaotic behaviour of the Solar System , 1989, Nature.

[18]  Martin J. Gander,et al.  On the Superlinear and Linear Convergence of the Parareal Algorithm , 2007, CSE 2007.

[19]  Bernard Philippe,et al.  A parallel shooting technique for solving dissipative ODE's , 1993, Computing.

[20]  E. Hairer.  Symmetric Projection Methods for Differential Equations on Manifolds , 2000 .

[21]  Martin J. Gander,et al.  Analysis of the Parareal Time-Parallel Time-Integration Method , 2007, SIAM J. Sci. Comput..

[22]  Guillaume Bal,et al.  On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations , 2005 .

[23]  Ernst Hairer,et al.  The life-span of backward error analysis for numerical integrators , 1997 .

[24]  Guillaume Bal,et al.  Symplectic parareal , 2006 .

[25]  Y. Maday,et al.  A “Parareal” Time Discretization for Non-Linear PDE’s with Application to the Pricing of an American Put , 2002 .

[26]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[27]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[28]  Kevin Burrage.  Parallel methods for ODEs , 1997, Adv. Comput. Math..

[29]  Patrick Joly.  The Mathematical Model for Elastic Wave Propagation , 2008 .

[30]  Kevin Burrage,et al.  Parallel and sequential methods for ordinary differential equations , 1995, Numerical analysis and scientific computation.

[31]  G. Benettin,et al.  On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms , 1994 .

[32]  G. Ciccotti,et al.  Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes , 1977 .

[33]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[34]  Einar M. Rønquist,et al.  Stability of the Parareal Algorithm , 2005 .

[35]  B. Leimkuhler,et al.  Symplectic Numerical Integrators in Constrained Hamiltonian Systems , 1994 .

[36]  Y Maday,et al.  Parallel-in-time molecular-dynamics simulations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Yvon Maday,et al.  The parareal in time algorithm , 2008 .

[38]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[39]  Izaskun Garrido Hernandez,et al.  Convergent iterative schemes for time parallelization , 2006, Math. Comput..

[40]  J. Laskar.  Chaotic diffusion in the Solar System , 2007, 0802.3371.

[41]  Alfredo Bellen,et al.  Parallel algorithms for initial-value problems for difference and differential equations , 1989 .