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On the time-parallelization of the solution of Navier-Stokes equations using Parareal Lunet, Thibaut


Unsteady turbulent flow simulations using the Navier Stokes equations require larger and larger problem sizes. On an other side, new supercomputer architectures will be available in the next decade, with computational power based on a larger number of cores rather than significantly increased CPU frequency. Hence most of the current generation CFD software will face critical efficiency issues if bounded to massive spatial parallelization and we consider time parallelization as an attractive alternative to enhance efficiency on multi- cores architectures. Several algorithms developed in the last decades (Parareal, PFASST) may be straightforwardly applied to the Navier-Stokes equations, but the Parareal algorithm remains one of the simplest solutions in the case of explicit time stepping, compressible flow Based on an optimized implementation of Parareal,3 we modelize the speed-up obtained when combining both space and time parallelizations. This modelization takes into account the speedup of an actual structured, massively parallel CFD solver and the cost of time communications, both measured on two different supercomputers. Some preliminary requirements for a worthy time-parallel integration will be then derived, in terms of both Parareal iteration count and size of the time subdomain window. We then study within this framework, possible enhancements of the well-known convergence difficulties for Parareal encountered for advection dominated problems. The proposed approach is based on the representation of Parareal as an algebraic system of nonlinear equations solved by a preconditioned Newton’s method. The new formulation targets the reduction of the degree of non-normality of its Jacobian by slightly modifying the Parareal iteration. Performance on examples related to canonical linear problems, like the Dahlquist and the one- dimensional advection equation, is analysed. To conclude we comment on the extension of this method to nonlinear problems.

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