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A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation

  • Special Issue Parallel-in-Time Methods
  • Published:
Computing and Visualization in Science

Abstract

This work focuses on the Parareal parallel-in-time method and its application to the viscous Burgers equation. A crucial component of Parareal is the coarse time stepping scheme, which strongly impacts the convergence of the parallel-in-time method. Three choices of coarse time stepping schemes are investigated in this work: explicit Runge–Kutta, implicit–explicit Runge–Kutta, and implicit Runge–Kutta with semi-Lagrangian advection. Manufactured solutions are used to conduct studies, which provide insight into the viability of each considered time stepping method for the coarse time step of Parareal. One of our main findings is the advantageous convergence behavior of the semi-Lagrangian scheme for advective flows.

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Authors and Affiliations

  1. Institute of Numerical Methods in Mechanical Engineering and Graduate School of Computational Engineering, TU Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    A. Schmitt & M. Schäfer

  2. University of Exeter, Harrison Building, North Park Road, Exeter, EX4 4QF, UK

    M. Schreiber

  3. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, Brazil

    P. Peixoto

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  1. A. Schmitt
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Correspondence to A. Schmitt.

Additional information

Schmitt: The work of this author is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt

Peixoto: Acknowledges the Sao Paulo Research Foundation (FAPESP) under the Grant Number 2016/18445-7 and the National Science and Technology Development Council (CNPq) under Grant Number 441328/2014-8.

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Schmitt, A., Schreiber, M., Peixoto, P. et al. A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation. Comput. Visual Sci. 19, 45–57 (2018). https://doi.org/10.1007/s00791-018-0294-1

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