A parallel space-time solver for the Navier-Stokes equations with periodic forcing

Prof. Dr. P. Arbenz, Computer Science Department, ETH Zurich
Prof. Dr. D. Obrist, ARTORG Center for Biomedical Engineering Research, University of Berne

Funding: This project is supported by the Swiss National Science Foundation by contract #200021_147052.


Modern solvers for the Navier-Stokes equations are optimized for massively-parallel computing platforms and can easily use many thousands of processing units by decomposing the spatial flow domain into small sub-domains. This allows nowadays the solution of complex unsteady flow problems at high Reynolds numbers with billions of grid points.

The continued increase in theoretical computational power should allow us to simulate ever larger flow problems as long as we are able to decompose the flow domain into sufficiently many sub-domains. The excellent weak scaling performance of modern solvers suggests that we should be able to scale the simulations without any limit in sight. However, this optimistic extrapolation does not factor in the increasing number of time steps which typically comes along with larger flow domains. This problem is aggravated for a common class of flow problems which are concerned with the spatial evolution of time-harmonic perturbations, e.g., a vibrating ribbon perturbing a boundary layer flow. The numerical simulation of such problems with classical time marching methods requires long transient periods at the beginning of each simulation. These transient periods are of no particular physical interest. The relevant flow field is only established when a periodic (or at least statistically steady) solution has been reached. In terms of dynamical systems, this can be understood as the asymptotic approach to a limit cycle of a given system.

For such problems with periodic forcing, the spatial decomposition of the computational domain might not yield sufficient parallelism for the efficient usage of modern massively-parallel supercomputers. Only a parallelization of the time integration could ease this potential scaling limit for flow problems of this type.

In the present project, we shall investigate whether the periodicity of the forcing and the existence of a limit cycle solution can be exploited by replacing the classical time marching scheme by a direct discretization of the limit cycle solution by a truncated Fourier series. This process leads to a very large nonlinear system of equations which has the set of all flow fields at every resolved time step (or temporal Fourier mode) as its unknowns. The efficient solution of such a large system of equations is major obstacle toward the simulation of complex flow problems (e.g. full airplane wings).

We aim at developing a solver for such a global problem which distributes the solution of the different Fourier modes to different processing units. Therefore, the time axis is parallelized such that the number of parallel threads scales with the number of resolved Fourier modes.


  1. D. Hupp, D. Obrist, P. Arbenz "A parallel Navier-Stokes solver using spectral discretization in time". Int. J. Comput. Fluid Dyn. 30 (7-10): 489-494 (2016), doi:10.1080/10618562.2016.1242725.
  2. P. Benedusi, D. Hupp, P. Arbenz, R. Krause: "A Parallel Multigrid Solver for Time-Periodic Incompressible Navier-Stokes Equations in 3D". In: Numerical Mathematics and Advanced Applications - ENUMATH 2015. B. Karasözen, M. Manguoglu, M. Tezer-Sezgin, S. Göktepe, Ö. Ugur (eds.). Lecture Notes in Computational Science and Engineering 112. Springer, 2016. pp. 265-273. doi:10.1007/978-3-319-39929-4_26
  3. D. Hupp, D. Obrist, P. Arbenz: "Multigrid preconditioning for time-periodic Navier-Stokes problems". Proc. Appl. Math. Mech. (PAMM) 15: 595-596 (2015), doi:10.1002/pamm.201510287
  4. P. Arbenz, D. Hupp, D. Obrist: "A parallel solver for the time-periodic Navier-Stokes Equations". In: Parallel Processing and Applied Mathematics (PPAM 13) Part II. R. Wyrzykowski, J. Dongarra, K. Karczewski, J. Waśniewski (eds.). Lecture Notes in Computer Science 8385, pp. 291-300. Springer, Berlin, 2014. doi:10.1007/978-3-642-55195-6_27.
  5. P. Arbenz, A. Hiltebrand, D. Obrist: "A parallel space-time finite difference solver for periodic solutions of the shallow-water equation". In: Parallel Processing and Applied Mathematics (PPAM 11) Part II. R. Wyrzykowski, J. Dongarra, K. Karczewski, J. Waśniewski (eds.). Lecture Notes in Computer Science 7204, pp.302-312. Springer, Berlin, 2012.
  6. D. Obrist, R. Henniger, P. Arbenz: "Parallelization of the time integration for time-periodic flow problems". Proc. Appl. Math. Mech. 10 (1): 567-568 (2010), doi:10.1002/pamm.201010276.


Prof. Dr. Peter Arbenz
ETH Zurich
Computer Science Department
Universitätsstrasse 6, CAB F51.1
CH-8092 Zürich
Tel.: +41 44 632 74 32
Email: arbenz@inf.ethz.ch

Prof. Dr. D. Obrist,
University of Bern
ARTORG Center for Biomedical Engineering Research, Cardiovascular Engineering
Murtenstrasse 50
CH-3008 Bern
Tel.: +41 31 632 7576
Email: dominik.obrist@artorg.unibe.ch

Last update: May 30 2017