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}
}
  • X. Dai, Y. Maday
  • Published 2013
  • Mathematics, Computer Science
  • SIAM J. Sci. Comput.
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
Analysis of Two Parareal Algorithms for Time-Periodic Problems
TLDR
A parareal algorithm with a periodic coarse problem and one with a nonperiodic coarse problem, which proves for both linear and nonlinear problems convergence. Expand
An efficient parareal algorithm for a class of time-dependent problems with fractional Laplacian
  • Shulin Wu
  • Mathematics, Computer Science
  • Appl. Math. Comput.
  • 2017
TLDR
A parareal algorithm for time-dependent diffusion equations with fractional Laplacian, which realizes parallel-in-time computation and provides a sharp estimate of the convergence rate, which is independent of the mesh ratio J and the distribution of the eigenvalues of the coefficient matrix. Expand
Multigrid Reduction in Time for non-linear hyperbolic equations
TLDR
This study aims at identifying the main causes for degradation in the convergence speed of the MGRIT algorithm, and finds the Courant-Friedrichs-Lewy (CFL) limit to be the principal determining factor. Expand
An Adjoint Approach for Stabilizing the Parareal Method
The parareal algorithm seeks to extract parallelism in the time-integration direction of time-dependent differential equations. While it has been applied with success to a wide range of problems, itExpand
Communication-aware adaptive Parareal with application to a nonlinear hyperbolic system of partial differential equations
TLDR
Trough large-scale numerical experiments it is demonstrated that with CAAP, it is possible not only to obtain time-parallel speedup on this class of evolution problems, but also that it may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. Expand
An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs
TLDR
A new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time of a highly oscillatory nature that combines the parareal method and techniques from the heterogeneous multiscale method is presented. Expand
A stable parareal-like method for the second order wave equation
TLDR
This work presents a data-driven strategy in which the computed data gathered from each iteration are re-used to stabilize the coupling by minimizing the wave energy residual of the fine and coarse propagated solutions. Expand
Optimizing MGRIT and Parareal coarse-grid operators for linear advection
TLDR
This paper applies MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically non-scalable or even divergent behavior of these solvers for this problem. Expand
On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method
We propose a modified parallel-in-time — parareal — multi-level time integration method that, in contrast to previously proposed methods, employs a coarse solver based on a reduced model, built fromExpand
Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions
We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method.Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 29 REFERENCES
The parareal in time algorithm
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 systemsExpand
A time-parallel implicit method for accelerating the solution of non-linear structural dynamics problems
TLDR
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
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 aExpand
A time-parallel algorithm for almost integrable Hamiltonian systems
TLDR
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
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 inExpand
Modified propagators of parareal in time algorithm and application to Princeton Ocean model
TLDR
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
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 styleExpand
Time‐decomposed parallel time‐integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications
TLDR
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
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
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. TheExpand
...
1
2
3
...