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

@article{Dai2013StablePI, title={Stable Parareal in Time Method for First- and Second-Order Hyperbolic Systems}, author={Xiaoying Dai and Yvon Maday}, journal={SIAM J. Sci. Comput.}, year={2013}, volume={35} }

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… Expand

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#### References

SHOWING 1-10 OF 29 REFERENCES

The parareal in time algorithm

- Mathematics
- 2008

In this paper we present the current status of a method, first introduced in 2001 authored by J.-L. Lions, Y. Maday and G. Turinici that allows for parallization in time for the simulation of systems… Expand

A time-parallel implicit method for accelerating the solution of non-linear structural dynamics problems

- Computer Science
- 2009

The parallel implicit time-integration algorithm (PITA) is extended to the non-linear case and its application to the reduction of the time-to-solution on a Linux cluster of sample non- linear structural dynamics problems is demonstrated. Expand

A parareal in time procedure for the control of partial differential equations

- Mathematics
- 2002

Abstract We have proposed in a previous note a time discretization for partial differential evolution equation that allows for parallel implementations. This scheme is here reinterpreted as a… Expand

A time-parallel algorithm for almost integrable Hamiltonian systems

- Computer Science, Mathematics
- ArXiv
- 2011

A time-parallel algorithm for solving numerically almost integrable Hamiltonian systems in action-angle coordinates that has a better convergence obtained from the use of derivatives of the perturbing term not considered in the original SST97 algorithm. Expand

Application of the Parareal Algorithm for Acoustic Wave Propagation

- Mathematics
- 2009

We present an application of the parareal algorithm [1] to solve wave propagation problems in the time domain. The parareal algorithm is based on a decomposition of the integration time interval in… Expand

Modified propagators of parareal in time algorithm and application to Princeton Ocean model

- Computer Science
- 2008

Two modified propagators of parareal in time algorithm are presented and applied to the Princeton Ocean model (POM) and the properties of the modified propagator are analyzed. Expand

Time-Dependent Problems and Difference Methods

- Mathematics
- 1996

Praise for the First Edition
". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style… Expand

Time‐decomposed parallel time‐integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications

- Computer Science
- 2003

This methodology parallelizes the time-loop of a time- dependent partial differential equation solver without interfering with its sequential or parallel space-computations for time-dependent problems with a few degrees of freedom such as those arising in robotics and protein folding applications. Expand

Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations

- Mathematics
- 2004

Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book.… Expand

Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses

- Engineering
- 2006

The time-parallel framework for constructing parallel implicit time-integration algorithms (PITA) is revisited in the specific context of linear structural dynamics and near-real-time computing. The… Expand