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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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References

  1. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997). https://doi.org/10.1016/S0168-9274(97)00056-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Baffico, L., Bernard, S., Maday, Y., Turinici, G., Zrah, G.: Parallel-in-time molecular-dynamics simulations. Phys. Rev. E 66(057), 701 (2002). https://doi.org/10.1103/PhysRevE.66.057701

    Google Scholar 

  3. Bal, G.: On the convergence and the stability of the parareal algorithm to solve partial differential equations. In: Kornhuber, R., et al. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 40, pp. 426–432. Springer, Berlin (2005). https://doi.org/10.1007/3-540-26825-1_43

    Chapter  Google Scholar 

  4. Barros, S.R., Peixoto, P.S.: Computational aspects of harmonic wavelet Galerkin methods and an application to a precipitation front propagation model. Comput. Math. Appl. 61(4), 1217–1227 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bateman, H.: Some recent researches on the motion of fluids. Mon. Weather Rev. 43(4), 163–170 (1915)

    Article  Google Scholar 

  6. Bauer, P., Thorpe, A., Brunet, G.: The quiet revolution of numerical weather prediction. Nature 525(7567), 47–55 (2015)

    Article  Google Scholar 

  7. Bonaventura, L.: An Introduction to Semi-Lagrangian Methods for Geophysical Scale Flows. Lecture Notes ERCOFTAC Leonhard Euler Lectures. SAM-ETH, Zurich (2004)

    Google Scholar 

  8. Burgers, J.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948). https://doi.org/10.1016/S0065-2156(08)70100-5

    Article  MathSciNet  Google Scholar 

  9. Christlieb, A.J., Macdonald, C.B., Ong, B.W.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010). https://doi.org/10.1137/09075740X

    Article  MathSciNet  MATH  Google Scholar 

  10. Côté, J.: Time-parallel algorithms for weather prediction and climate simulation. https://www.newton.ac.uk/files/seminar/20121023121012359-153396.pdf. Accessed 03 May 2017 (2012)

  11. Durran, D.: Numerical Methods for Fluid Dynamics: With Applications to Geophysics. Texts in Applied Mathematics. Springer, New York (2010). https://doi.org/10.1007/978-1-4419-6412-0

    Book  MATH  Google Scholar 

  12. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241–266 (2000). https://doi.org/10.1023/A:1022338906936

    Article  MathSciNet  MATH  Google Scholar 

  13. Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7, 105–132 (2012). https://doi.org/10.2140/camcos.2012.7.105

    Article  MathSciNet  MATH  Google Scholar 

  14. Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators: Theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Method Eng. 58(9), 1397–1434 (2003). https://doi.org/10.1002/nme.860

    Article  MathSciNet  MATH  Google Scholar 

  15. Friedhoff, S., Falgout, R.D., Kolev, T.V., MacLachlan, S.P., Schroder, J.B.: A multigrid-in-time algorithm for solving evolution equations in parallel. In: Presented at Sixteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, United States, Mar 17–22, 2013, http://www.osti.gov/scitech/servlets/purl/1073108 (2013)

  16. Gander, M.J.: Analysis of the parareal algorithm applied to hyperbolic problems using characteristics. Boletin de la Sociedad Espanola de Matemática Aplicada 42, 21–35 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23321-5_3

    Google Scholar 

  18. Gander, M.J., Hairer, E.: Analysis for parareal algorithms applied to Hamiltonian differential equations. J. Comput. Appl. Math. 259, Part A(0):2–13. In: Proceedings of the Sixteenth International Congress on Computational and Applied Mathematics (ICCAM-2012), Ghent, Belgium, 9–13 July, 2012 (2014). https://doi.org/10.1016/j.cam.2013.01.011

  19. Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007). https://doi.org/10.1137/05064607X

    Article  MathSciNet  MATH  Google Scholar 

  20. Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1977)

    Book  MATH  Google Scholar 

  21. Hortal, M.: The development and testing of a new two-time-level semi-lagrangian scheme (SETTLS) in the ECMWF forecast model. Q. J. R. Meteorol. Soc. 128(583), 1671–1687 (2002). https://doi.org/10.1002/qj.200212858314

    Article  Google Scholar 

  22. Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16(4), 848–864 (1995). https://doi.org/10.1137/0916050

    Article  MathSciNet  MATH  Google Scholar 

  23. Horton, G., Vandewalle, S., Worley, P.: An algorithm with polylog parallel complexity for solving parabolic partial differential equations. SIAM J. Sci. Comput. 16(3), 531–541 (1995). https://doi.org/10.1137/0916034

    Article  MathSciNet  MATH  Google Scholar 

  24. Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1), 139–181 (2003). https://doi.org/10.1016/S0168-9274(02)00138-1

    Article  MathSciNet  MATH  Google Scholar 

  25. Kooij, G.L., Botchev, M.A., Geurts, B.J.: A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. J. Comput. Appl. Math. 316, 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036. (Selected Papers from NUMDIFF-14)

    Article  MathSciNet  MATH  Google Scholar 

  26. LeVeque, R.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM e-books, Society for Industrial and Applied Mathematics, Philadelphia (2007)

    Book  MATH  Google Scholar 

  27. Lions, J.L., Maday, Y., Turinici, G.: A parareal in time discretization of PDE’s. Comptes Rendus de l’Académie des Sci.-Ser. I—Math. 332, 661–668 (2001). https://doi.org/10.1016/S0764-4442(00)01793-6

    MATH  Google Scholar 

  28. Lubich, C., Ostermann, A.: Multi-grid dynamic iteration for parabolic equations. BIT Numer. Math. 27(2), 216–234 (1987). https://doi.org/10.1007/BF01934186

    Article  MATH  Google Scholar 

  29. Peixoto, P.S., Barros, S.R.: On vector field reconstructions for semi-Lagrangian transport methods on geodesic staggered grids. J. Comput. Phys. 273, 185–211 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., et al.: Numerical Recipes, vol. 3. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  31. Reynolds-Barredo, J., Newman, D., Sanchez, R., Jenko, F.: A novel, semilagrangian, coarse solver for the Parareal technique and its application to 2D fluid drift-wave (BETA) and 5D gyrokinetic (GENE), turbulence simulations. In: APS Meeting Abstracts, vol 1, p 8128P (2013)

  32. Ruprecht, D., Krause, R.: Explicit parallel-in-time integration of a linear acoustic-advection system. Comput. Fluids 59, 72–83 (2012). https://doi.org/10.1016/j.compfluid.2012.02.015

    Article  MathSciNet  MATH  Google Scholar 

  33. Schreiber, M., Peixoto, P.S., Haut, T., Wingate, B.: Beyond spatial scalability limitations with a massively parallel method for linear oscillatory problems. Int. J. High Perform. Comput. Appl. 29(3), 261–273 (2016)

    Google Scholar 

  34. Schreiber, M., Peixoto, P., Schmitt, A.: (2017) https://github.com/schreiberx/sweet. Accessed 4 May 2017

  35. Staff, G.A., Rønquist, E.M.: Stability of the parareal algorithm. In: Kornhuber, R., et al. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 40, pp. 449–456. Springer, Berlin (2005). https://doi.org/10.1007/3-540-26825-1_46

    Chapter  Google Scholar 

  36. Staniforth, A., Côté, J.: Semi-Lagrangian integration schemes for atmospheric models—a review. Mon. Weather Rev. 119(9), 2206–2223 (1991)

    Article  Google Scholar 

  37. Steiner, J., Ruprecht, D., Speck, R., Krause, R.: Convergence of Parareal for the Navier–Stokes Equations Depending on the Reynolds Number, pp. 195–202. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-10705-9_19

    MATH  Google Scholar 

  38. Sutter, H.: The free lunch is over: a fundamental turn toward concurrency in software. Dr Dobbs J. 30(3), 202–210 (2005)

    Google Scholar 

  39. Swarztrauber, P.N., Sweet, R.A.: The Fourier and Cyclic Reduction Methods for Solving Poissons Equation. Wiley, New York (1996)

    Google Scholar 

  40. Vandewalle, S., Van de Velde, E.: Space-time concurrent multigrid waveform relaxation. Ann. Numer. Math. 1(1–4), 347–360 (1994). https://doi.org/10.13140/2.1.1146.1761

    MathSciNet  MATH  Google Scholar 

  41. Wedi, N.P., Bauer, P., Deconinck, W., Diamantakis, M., Hamrud, M., Kuehnlein, C., Malardel, S., Mogensen, K., Mozdzynski, G., Smolarkiewicz, P.K.: The modelling infrastructure of the integrated forecasting system: recent advances and future challenges. European Centre for Medium-Range Weather Forecasts (2015)

  42. Wesseling, P.: Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics. Springer, Berlin (2009)

    MATH  Google Scholar 

  43. Williamson, D.L.: The evolution of dynamical cores for global atmospheric models. J. Meteorol. Soc. Japan Ser. II 85, 241–269 (2007)

    Article  Google Scholar 

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Correspondence to A. Schmitt.

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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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