Abstract
In this paper, we will report our recent effort to apply the parareal algorithm to the time parallelization of an industrial code that simulates two phase flows in a reactor for safety studies. This software solves the six equation two-fluid model by considering a set of balance laws (mass, momentum and energy) for each phase, liquid and vapor, of the fluid. The discretization is based on a finite volume method on a staggered grid in space and on a multistep time scheme. Here, we apply a variant of the parareal algorithm on an oscillating manometer test case: the multistep variant allowing to deal with multistep time schemes in the coarse and/or fine propagators. Numerical results show that parareal methods offer the potential for an increased level of parallelism and is a good strategy to complement the current space domain decomposition implemented in the code.
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References
Ait-Ameur, K., Maday, Y., Tajchman, M.: Multi-step variant of the parareal algorithm. In: Haynes, R.D., MacLachlan, S., Cai, X.C., Halpern, L., Kim, H.H., Klawonn, A., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XXV, Lecture Notes in Computational Science and Engineering, pp. 393–400 (2020)
Audouze, C., Massot, M., Volz, S.: Symplectic multi-time step parareal algorithms applied to molecular dynamics. http://hal.archives-ouvertes.fr/hal-00358459/fr/ (2009)
Baffico, L., Bernard, S., Maday, Y., Turinici, G., Zrah, G.: Parallel-in-time molecular dynamics simulations. Phys. Rev. E 66, 057701 (2002)
Bal, G.: Parallelization in time of (stochastic) ordinary differential equations. http://www.columbia.edu/gb2030/PAPERS/paralleltime.pdf (2003)
Bal, G., Maday, Y.: A “parareal” time discretization for non-linear PDE’s with application to the pricing of an American put. In: Recent Developments in Domain Decomposition Methods, vol. 23, pp. 189–202 (2002)
Dellacherie, S.: Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys. 229(4), 978–1016 (2010)
Drew, D.A., Passman, S.L.: Theory of Multicomponent Fluids. Springer, New York (1999)
Fischer, P.F., Hecht, F., Maday, Y.: A parareal in time semi-implicit approximation of the Navier-Stokes equations. In: Kornhuber, R., et al. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 40, pp. 433–440, Springer, Berlin (2004)
Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Krkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition Methods, pp. 69–114. Springer, Cham (2015)
Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)
Hewitt, G.F., Delhaye, J.M., Zuber, N.: Multiphase Science and Technology, vol. 6. Springer, New York (1991)
M. Ishii, Thermo-fluid Dynamic Theory of Two-phase Flow. Eyrolles, Paris (1975)
Lions, J.-L., Maday, Y., Turinici, G.: Résolution par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris 332(7), 661–668 (2001)
Maday, Y., Mula, O.: An adaptive parareal algorithm. J. Comput. Appl. Math. (2020). https://arxiv.org/pdf/1909.08333.pdf
Ndjinga, M.: Influence of interfacial pressure on the hyperbolicity of the two-fluid model. C. R. Acad. Sci. Paris Ser. I 344, 407–412 (2007)
Ndjinga, M., Nguyen, T.P.K., Chalons, C.: A 2 × 2 hyperbolic system modelling incompressible two phase flows: theory and numerics. Nonlinear Differ. Equ. Appl. 24, 36 (2017). https://doi.org/10.1007/s00030-017-0458-6
Samaddar, D., Newman, D.E., Sanchez, R.: Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm. J. Comput. Phys. 229(18), 6558–6573 (2010)
Staff, G.A., Ronquist, E.M.: Stability of the Parareal algorithm, In: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 40, pp. 449–456. Springer, Berlin (2005)
Acknowledgements
This research is partially supported by ANR project CINE-PARA (ANR-15-CE23-0019). We acknowledge the travel support provided by SMAI to attend the Spanish-French School Jacques-Louis Lions.
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Ait-Ameur, K., Maday, Y., Tajchman, M. (2021). Time-Parallel Algorithm for Two Phase Flows Simulation. In: Greiner, D., Asensio, M.I., Montenegro, R. (eds) Numerical Simulation in Physics and Engineering: Trends and Applications. SEMA SIMAI Springer Series, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-030-62543-6_5
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