Close Search Advanced
Journals
Resources
About Us
Open Access

The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations

The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations
Preview
Add to basket

Year:    2010

Journal of Computational Mathematics, Vol. 28 (2010), Iss. 5 : pp. 676–692

Abstract

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Offline-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1003-m2980

Journal of Computational Mathematics, Vol. 28 (2010), Iss. 5 : pp. 676–692

Published online:    2010-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

Keywords:    Finite element and spectral element approximations Multi-meshes and multi-degrees techniques Reduced basis technique Semi-implicit Runge-Kutta scheme Offline-online procedure Parareal in time algorithm.

  1. Analysis of a fractional-step parareal algorithm for the incompressible Navier-Stokes equations

    Miao, Zhen | Zhang, Ren-Hao | Han, Wei-Wei | Jiang, Yao-Lin

    Computers & Mathematics with Applications, Vol. 161 (2024), Iss. P.78

    https://doi.org/10.1016/j.camwa.2024.02.035 [Citations: 0]

  2. Extending molecular simulation time scales: Parallel in time integrations for high-level quantum chemistry and complex force representations

    Bylaska, Eric J. | Weare, Jonathan Q. | Weare, John H.

    The Journal of Chemical Physics, Vol. 139 (2013), Iss. 7

    https://doi.org/10.1063/1.4818328 [Citations: 18]

  3. Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows

    Miao, Zhen | Jiang, Yao-Lin | Yang, Yun-Bo

    International Journal of Computer Mathematics, Vol. 96 (2019), Iss. 7 P.1398

    https://doi.org/10.1080/00207160.2018.1498484 [Citations: 8]
  4. Reduced Order Methods for Modeling and Computational Reduction

    On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

    Chen, Feng | Hesthaven, Jan S. | Zhu, Xueyu

    2014

    https://doi.org/10.1007/978-3-319-02090-7_7 [Citations: 15]

  5. Parareal in Time Simulation Of Morphological Transformation in Cubic Alloys with Spatially Dependent Composition

    He, Li-Ping | He, Minxin

    Communications in Computational Physics, Vol. 11 (2012), Iss. 5 P.1697

    https://doi.org/10.4208/cicp.110310.090911a [Citations: 6]

  6. Parareal Convergence for Oscillatory PDEs with Finite Time-Scale Separation

    Peddle, Adam G. | Haut, Terry | Wingate, Beth

    SIAM Journal on Scientific Computing, Vol. 41 (2019), Iss. 6 P.A3476

    https://doi.org/10.1137/17M1131611 [Citations: 9]

  7. A Time-Parallel Framework for Coupling Finite Element and Lattice Boltzmann Methods

    Astorino, Matteo | Chouly, Franz | Quarteroni, Alfio

    Applied Mathematics Research eXpress, Vol. 2016 (2016), Iss. 1 P.24

    https://doi.org/10.1093/amrx/abv009 [Citations: 5]

  8. Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

    Baudron, Anne-Marie | Lautard, Jean-Jacques | Maday, Yvon | Riahi, Mohamed Kamel | Salomon, Julien

    Journal of Computational Physics, Vol. 279 (2014), Iss. P.67

    https://doi.org/10.1016/j.jcp.2014.08.037 [Citations: 14]

  9. Parareal computation of stochastic differential equations with time-scale separation: a numerical convergence study

    Legoll, Frédéric | Lelièvre, Tony | Myerscough, Keith | Samaey, Giovanni

    Computing and Visualization in Science, Vol. 23 (2020), Iss. 1-4

    https://doi.org/10.1007/s00791-020-00329-y [Citations: 5]

  10. Parareal Multiscale Methods for Highly Oscillatory Dynamical Systems

    Ariel, Gil | Kim, Seong Jun | Tsai, Richard

    SIAM Journal on Scientific Computing, Vol. 38 (2016), Iss. 6 P.A3540

    https://doi.org/10.1137/15M1011044 [Citations: 12]

  11. A New Parareal Algorithm for Problems with Discontinuous Sources

    Gander, Martin J. | Kulchytska-Ruchka, Iryna | Niyonzima, Innocent | Schöps, Sebastian

    SIAM Journal on Scientific Computing, Vol. 41 (2019), Iss. 2 P.B375

    https://doi.org/10.1137/18M1175653 [Citations: 19]

  12. Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics

    Blumers, Ansel L. | Li, Zhen | Karniadakis, George Em

    Journal of Computational Physics, Vol. 393 (2019), Iss. P.214

    https://doi.org/10.1016/j.jcp.2019.05.016 [Citations: 8]