Communications in Applied Mathematics and Computational Science

Toward an efficient parallel in time method for partial differential equations

Matthew Emmett and Michael Minion

Full-text: Open access

Abstract

A new method for the parallelization of numerical methods for partial differential equations (PDEs) in the temporal direction is presented. The method is iterative with each iteration consisting of deferred correction sweeps performed alternately on fine and coarse space-time discretizations. The coarse grid problems are formulated using a space-time analog of the full approximation scheme popular in multigrid methods for nonlinear equations. The current approach is intended to provide an additional avenue for parallelization for PDE simulations that are already saturated in the spatial dimensions. Numerical results and timings on PDEs in one, two, and three space dimensions demonstrate the potential for the approach to provide efficient parallelization in the temporal direction.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 7, Number 1 (2012), 105-132.

Dates
Received: 21 December 2011
Revised: 18 January 2012
Accepted: 29 January 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513732042

Digital Object Identifier
doi:10.2140/camcos.2012.7.105

Mathematical Reviews number (MathSciNet)
MR2979518

Zentralblatt MATH identifier
1248.65106

Subjects
Primary: 65M99: None of the above, but in this section

Keywords
parallel computing time parallel ordinary differential equations partial differential equations deferred corrections parareal

Citation

Emmett, Matthew; Minion, Michael. Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7 (2012), no. 1, 105--132. doi:10.2140/camcos.2012.7.105. https://projecteuclid.org/euclid.camcos/1513732042


Export citation

References

  • U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), no. 2-3, 151–167.
  • G. Bal, On the convergence and the stability of the parareal algorithm to solve partial differential equations, Domain decomposition methods in science and engineering (R. Kornhuber et al., eds.), Lect. Notes Comput. Sci. Eng., no. 40, Springer, Berlin, 2005, pp. 425–432.
  • G. Bal and Y. Maday, A “parareal” time discretization for non-linear PDE's with application to the pricing of an American put, Recent developments in domain decomposition methods (L. F. Pavarino and A. Toselli, eds.), Lect. Notes Comput. Sci. Eng., no. 23, Springer, Berlin, 2002, pp. 189–202.
  • K. B öhmer, P. Hemker, and H. J. Stetter, The defect correction approach, Defect correction methods (K. Böhmer and H. J. Stetter, eds.), Comput. Suppl., no. 5, Springer, Vienna, 1984, pp. 1–32.
  • S. Boscarino, Error analysis of IMEX Runge–Kutta methods derived from differential-algebraic systems, SIAM J. Numer. Anal. 45 (2007), no. 4, 1600–1621.
  • A. Bourlioux, A. T. Layton, and M. L. Minion, High-order multi-implicit spectral deferred correction methods for problems of reactive flow, J. Comput. Phys. 189 (2003), no. 2, 651–675.
  • E. L. Bouzarth and M. L. Minion, A multirate time integrator for regularized Stokeslets, J. Comput. Phys. 229 (2010), no. 11, 4208–4224.
  • W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
  • J. W. Daniel, V. Pereyra, and L. L. Schumaker, Iterated deferred corrections for initial value problems, Acta Ci. Venezolana 19 (1968), 128–135.
  • A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), no. 2, 241–266.
  • C. Farhat and M. Chandesris, Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications, Internat. J. Numer. Methods Engrg. 58 (2003), no. 9, 1397–1434.
  • P. F. Fischer, F. Hecht, and Y. Maday, A parareal in time semi-implicit approximation of the Navier–Stokes equations, Domain decomposition methods in science and engineering (R. Kornhuber et al., eds.), Lect. Notes Comput. Sci. Eng., no. 40, Springer, Berlin, 2005, pp. 433–440.
  • M. J. Gander, Analysis of the parareal algorithm applied to hyperbolic problems using characteristics, Bol. Soc. Esp. Mat. Apl. S$\vec{\rm e}$MA (2008), no. 42, 21–35.
  • J. Huang, J. Jia, and M. Minion, Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys. 214 (2006), no. 2, 633–656.
  • C. A. Kennedy and M. H. Carpenter, Additive Runge–Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math. 44 (2003), no. 1-2, 139–181.
  • I. G. Kevrekidis, B. Nicolaenko, and J. C. Scovel, Back in the saddle again: a computer assisted study of the Kuramoto–Sivashinsky equation, SIAM J. Appl. Math. 50 (1990), no. 3, 760–790.
  • A. T. Layton, On the choice of correctors for semi-implicit Picard deferred correction methods, Appl. Numer. Math. 58 (2008), no. 6, 845–858.
  • A. T. Layton and M. L. Minion, Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics, J. Comput. Phys. 194 (2004), no. 2, 697–715.
  • ––––, Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods, Commun. Appl. Math. Comput. Sci. 2 (2007), 1–34.
  • J.-L. Lions, Y. Maday, and G. Turinici, Résolution d'EDP par un schéma en temps “pararéel”, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 7, 661–668.
  • M. L. Minion and S. A. Williams, Parareal and spectral deferred corrections, AIP Conference Proceedings, vol. 1048, 2008, pp. 388–391.
  • M. L. Minion, Higher-order semi-implicit projection methods, Numerical simulations of incompressible flows (M. Hafez, ed.), World Sci. Publ., River Edge, NJ, 2003, pp. 126–140.
  • ––––, Semi-implicit projection methods for incompressible flow based on spectral deferred corrections, Appl. Numer. Math. 48 (2004), no. 3-4, 369–387.
  • ––––, A hybrid parareal spectral deferred corrections method, Commun. Appl. Math. Comput. Sci. 5 (2010), no. 2, 265–301.
  • L. Pareschi and G. Russo, Implicit-explicit Runge–Kutta schemes for stiff systems of differential equations, Recent trends in numerical analysis (D. Trigiante, ed.), Adv. Theory Comput. Math., no. 3, Nova Scientific, Huntington, NY, 2001, pp. 269–288.
  • V. Pereyra, On improving an approximate solution of a functional equation by deferred corrections, Numer. Math. 8 (1966), 376–391.
  • ––––, Iterated deferred corrections for nonlinear operator equations, Numer. Math. 10 (1967), 316–323.
  • J. W. Shen and X. Zhong, Semi-implicit Runge–Kutta schemes for the non-autonomous differential equations in reactive flow computations, Proceedings of the 27th AIAA Fluid Dynamics Conference, AIAA, June 1996, pp. 17–20.
  • H. J. Stetter, Economical global error estimation, Stiff differential systems (R. A. Willoughby, ed.), Plenum, New York, 1974, pp. 245–258.
  • P. Zadunaisky, A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations, The Theory of Orbits in the Solar System and in Stellar Systems. Proceedings of International Astronomical Union, Symposium 25 (G. Contopoulos, ed.), 1964, pp. 281–287.