TU CE GSC CE People at the Graduate School CE Students Andreas Schmitt Research Project of Andreas Schmitt

Numerical investigation of Parallel-in-Time methods for dominantly hyperbolic equations

The increase in computational power is mostly due to larger amount of processing units in a supercomputer and not due to the increased clock speed of these units. Therefore it is necessary to use a parallel solver such that the work of the computation of the problem can be distributed to multiple processing units.

Nowadays, the standard is to apply parallelization in space by dividing the computational domain in multiple sub domains which get each solved on one unit. Between these sub domains the boundary information has to be exchanged so that the parallelized version calculates the same result a serial version would attain. Due to this communication overhead it is not possible to utilize an arbitrary amount of processing units efficiently. This limit may be reached before all available processing units of a supercomputer are utilized.

A fairly recent approach to tackle this utilization problem are the so-called parallel-in-time methods. As the name suggests, the idea is to use vacant cores for a parallelization of the time domain. First ideas for these methods started as early as 1964 by Nievergelt [3]. The most known method is the Parareal method proposed by Lions et al. in 2001 [2]. Further algorithms can be found e.g. in the review article by Gander [1].

The core of this project is to investigate the efficiency of a parallel in time algorithm applied to problems of hyperbolic, or nearly hyperbolic nature. For this the viscous Burgers equation is used with varying viscosities. The motivation lies on gaining speed up for incompressible flow problems beyond the saturation of the spacial parallelization.

[1] M. J. Gander. 50 years of Time Parallel Time Integration. In *Multiple Shooting and Time Domain Decomposition*. Springer 2015.

[2] J.-L. Lions, Y. Maday, and G. Turinici. A "parareal" in time discretization of PDE's. *Comptes Rendus de l'Académie des Sciences - Series I - Mathematics*, 332:661-668, 2001.

[3] J. Nievergelt. Parallel methods for integrating ordinary differential equations. *Commun. ACM*, 7(12):731-733, 1964.

K1) Modeling and simulation of coupled multi-physical problems

R7) Multiphysics applications

C2) High-performance computing techniques

Andreas Schmitt,

M.Sc.

Address: | Dolivostraße 15 |

D-64293 Darmstadt | |

Germany | |

Phone: | +49 6151 16 - 24377 |

Fax: | +49 6151 16 - 24404 |

Office: | S4|10-312 |

Email: | |