Convergence of Parareal for the Navier-Stokes Equations Depending on the Reynolds Number
The paper presents first a linear stability analysis for the time-parallel Parareal method, using an IMEX Euler as coarse and a Runge-Kutta-3 method as fine propagator, confirming that dominant imaginary eigenvalues negatively affect Parareal’s convergence. This suggests that when Parareal is applied to the nonlinear Navier-Stokes equations, problems for small viscosities could arise. Numerical results for a driven cavity benchmark are presented, confirming that Parareal’s convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection. The effect is found to strongly depend on the spatial resolution.
KeywordsReynolds Number Linear Stability Analysis Stability Domain Domain Decomposition Method Drive Cavity
Unable to display preview. Download preview PDF.
- 1.G. Bal, On the convergence and the stability of the parareal algorithm to solve partial differential equations, in Domain Decomposition Methods in Science and Engineering, ed. by R. Kornhuber et al. Lecture Notes in Computational Science and Engineering, vol. 40 (Springer, Berlin, 2005), pp. 426–432Google Scholar
- 3.F. Chen, J. Hesthaven, X. Zhu, On the use of reduced basis methods to accelerate and stabilize the parareal method, in Reduced Order Methods for Modeling and Computational Reduction. MS&A – Modeling, Simulation and Applications, vol. 9 (Springer, International Publishing Switzerland, 2014)Google Scholar
- 5.R. Croce, D. Ruprecht, R. Krause, Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow. Modeling, Simulation and Optimization of Complex Processes (Springer, Berlin/Heidelberg, 2012, in press)Google Scholar
- 10.P.F. Fischer, F. Hecht, Y. Maday, A parareal in time semi-implicit approximation of the Navier-Stokes equations, in Domain Decomposition Methods in Science and Engineering, ed. by R. Kornhuber et al. Lecture Notes in Computational Science and Engineering, vol. 40 (Springer, Berlin, 2005), pp. 433–440Google Scholar
- 13.J.-L. Lions, Y. Maday, G. Turinici, A “parareal” in time discretization of PDE’s, Comptes Rendus de l’Académie des Sciences – Series I – Mathematics 332, 661–668 (2001)Google Scholar
- 15.R. Speck, D. Ruprecht, R. Krause, M. Emmett, M. Minion, M. Winkel, P. Gibbon, A massively space-time parallel n-body solver, in Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, Salt Lake City (IEEE Computer Society Press, Los Alamitos, 2012), pp. 92:1–92:11Google Scholar
- 16.G.A. Staff, E.M. Rønquist, Stability of the parareal algorithm, in Domain Decomposition Methods in Science and Engineering ed. by R. Kornhuber et al. Lecture Notes in Computational Science and Engineering, vol. 40 (Springer, Berlin, 2005), pp. 449–456Google Scholar
- 18._____________________________________________ , Parallel-in-time simulation of two-dimensional, unsteady, incompressible laminar flows. Numer. Heat Transf. Part B: Fundam. 50(1), 25–40 (2006)Google Scholar
- 19.H. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method (Pearson Education, Harlow, England, 2007)Google Scholar