# Parareal for Diffusion Problems with Space- and Time-Dependent Coefficients

## Abstract

The very rapidly increasing number of cores in state-of-the-art supercomputers fuels both the need for and the interest in novel numerical algorithms inherently designed to feature concurrency. In addition to the mature field of space-parallel approaches (e.g. domain decomposition techniques), time-parallel methods that allow concurrency along the temporal dimension are now an increasingly active field of research, although first ideas, like in [12], go back several decades. A prominent and widely studied algorithm in this area is Parareal, introduced in [10], which has the advantage that one can couple and reuse classical time-stepping schemes in an iterative fashion to parallelize in time. However, there also exist a number of other approaches, e.g. the “parallel implicit time algorithm” (PITA) from [5], the “parallel full approximation scheme in space and time” (PFASST) from [4] or “revisionist integral deferred corrections” (RIDC) from [3] to name a few. Parareal in particular and temporal parallelism in general has been considered early as an addition to spatial parallelism in order to extend strong scaling limits, see [11]. Efficacy of this approach in large-scale parallel simulations on hundreds of thousands of cores has been demonstrated for the PFASST algorithm in [14].

## Keywords

Parareal Iteration Iteration Matrix Fixed Point Iteration Implicit Euler Method Domain Decomposition Technique## Notes

### Acknowledgements

This work was supported by the Swiss National Science Foundation (SNSF) under the lead agency agreement through the project “ExaSolvers” within the Priority Programme 1648 “Software for Exascale Computing” (SPPEXA) of the Deutsche Forschungsgemeinschaft (DFG). The authors thankfully acknowledge support from Achim Schädle, who provided parts of the used code.

## References

- 1.P. Amodio, L. Brugnano, Parallel solution in time of odes: some achievements and perspectives. Appl. Numer. Math.
**59**(3–4), 424–435 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 2.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.A.J. Christlieb, C.B. Macdonald, B.W. Ong, Parallel high-order integrators. SIAM J. Sci. Comput.
**32**(2), 818–835 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 4.M. Emmett, M.L. Minion, Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci.
**7**, 105–132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 5.C. Farhat, M. Chandesris, Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Methods Eng.
**58**(9), 1397–1434 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 6.S. Friedhoff, R.D. Falgout, T.V. Kolev, S. MacLachlan, J.B. Schroder, A multigrid-in-time algorithm for solving evolution equations in parallel, in
*Sixteenth Copper Mountain Conference on Multigrid Methods*, Copper Mountain, 17–22 March 2013Google Scholar - 7.M. Gander, E. Hairer, Nonlinear convergence analysis for the parareal algorithm, in
*Domain Decomposition Methods in Science and Engineering*, ed. by U. Langer, O. Widlund, D. Keyes. Lecture Notes in Computational Science and Engineering, vol. 60 (Springer, Berlin/Heidelberg, 2008), pp. 45–56Google Scholar - 8.M. Gander, M. Petcu, Analysis of a Krylov subspace enhanced parareal algorithm for linear problems. ESAIM Proc.
**25**, 114–129 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 9.M.J. Gander, S. Vandewalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput.
**29**(2), 556–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 10.J.-L. Lions, Y. Maday, G. Turinici, A “parareal” in time discretization of PDE’s. C. R. Acad. Sci. Ser. I Math.
**332**, 661–668 (2001)MathSciNetzbMATHGoogle Scholar - 11.Y. Maday, G. Turinici, The parareal in time iterative solver: a further direction to parallel implementation, 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. 441–448Google Scholar - 12.J. Nievergelt, Parallel methods for integrating ordinary differential equations. Commun. ACM
**7**(12), 731–733 (1964)MathSciNetCrossRefzbMATHGoogle Scholar - 13.D. Ruprecht, R. Krause, Explicit parallel-in-time integration of a linear acoustic-advection system. Comput. Fluids
**59**, 72–83 (2012)MathSciNetCrossRefGoogle Scholar - 14.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*, SC ’12 (IEEE Computer Society, Los Alamitos, 2012), pp. 92:1–92:11Google Scholar