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On the Efficiency of the Parallel-in-Time Finite Volume Calculation of the Unsteady Navier-Stokes Equations

J. M. F. Trindade1, J. C. F. Pereira2

Escola Náutica Infante D. Henrique, Av. Bonneville Franco, 2780-572 Pa co d’Arcos, Portugal.(
Instituto Superior Técnico, Universidade Técnica de Lis-boa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. (

Computer Modeling in Engineering & Sciences 2007, 20(1), 1-10.


In this paper, we discuss the efficiency and speed-up of parallel-in-time calculations of the unsteady incompressible Navier-Stokes equations in a PC-cluster. The parallel-in-time method is based on the alternate use of coarse global sequential solvers with fine local parallel ones in an iterative predictor-corrector fashion. Therefore, the efficiency of parallel calculations is strongly dependent on the number of iterations required for convergence. The one-dimensional scalar transport equation and the two-dimensional incompressible unsteady form of the Navier-Stokes equations were used to conduct numerical experiments to derive some conclusions concerning the accuracy and convergence of the iterative method. A simple performance model is proposed to estimate the efficiency of the parallel calculations as a function of the most relevant parameters that contribute to the computing time required to perform a parallel-in-time calculation. Among them, we have analyzed the influence of the number of processors, the number of iterations in the parallel-in-time algorithm and the influence of the coarse to fine time-grid step size ratio. The good agreement between the obtained parallel efficiency and the values estimated by the proposed performance model allows to conclude that parallel-in-time efficiency is quantitatively different from the parallel efficiency of the space domain decomposition, regarding the number of processors available for a fixed problem dimension. A significant speed-up is possible when the temporal scale of the problem is large and enough processors are available.


Cite This Article

M., J., C., J. (2007). On the Efficiency of the Parallel-in-Time Finite Volume Calculation of the Unsteady Navier-Stokes Equations. CMES-Computer Modeling in Engineering & Sciences, 20(1), 1–10.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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