# Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow

@article{Croce2012ParallelinSpaceandTimeSO, title={Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow}, author={Roberto Croce and Daniel Ruprecht and Rolf H. Krause}, journal={ArXiv}, year={2012}, volume={abs/1705.06149} }

In this paper we combine the Parareal parallel-in-time method together with spatial parallelization and investigate this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations. Parallelization of time stepping provides a new direction of parallelization and allows to employ additional cores to further speed up simulations after spatial parallelization has saturated. We report on numerical experiments performed on a Cray XE6, simulating a…

## 22 Citations

### Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening

- Computer ScienceComput. Vis. Sci.
- 2018

It is shown that a single Parareal iteration is able to reproduce with high fidelity the main statistical quantities characterizing the turbulent flow field with respect to speedup and quality of the solution.

### Parallel time-stepping for fluid-structure interactions

- Computer ScienceMathematical Modelling of Natural Phenomena
- 2021

It turns out that especially processes based on an internal dynamics cause great difficulties, but configurations which are driven by oscillatory problem data are well-suited for parallel time stepping and allow for substantial speedups.

### Space-time block preconditioning for incompressible flow

- Computer ScienceSIAM J. Sci. Comput.
- 2022

This paper develops a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and aspace-time matrix-vector multiplication.

### Hybrid Space-Time Parallel Solution of Burgers Equation

- Computer Science
- 2014

An OpenMP-based shared memory implementation of the Parareal parallel-in-time integration scheme using explicit integrators and a standard MPI-based spatial parallelization of a finite difference method into a hybrid space–time parallel scheme to achieve speedups beyond the saturation of the pure space-parallel scheme.

### Space-Time Parallelization of Fluid Dynamics Simulations

- Computer Science
- 2018

The parareal algorithm is one of the strongest candidates in the search for parallelization of the temporal dimension of time dependent problems and has been employed in time-parallel simulation of simple test problems.

### Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows

- Computer ScienceInt. J. Comput. Math.
- 2019

A parareal-in-time scheme for the incompressible non-isothermal Navier–Stokes equations with Boussinesq approximation is presented and it theoretically demonstrates the superlinearly convergence of theParareal iteration combined with finite element method for incompressable non- isothermal flows.

### Computational Physics: X Parallel-in-time integration of kinematic dynamos

- Physics
- 2020

The precise mechanisms responsible for the natural dynamos in the Earth and Sun are still not fully understood. Numerical simulations of natural dynamos are extremely computationally intensive, and…

### Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number

- MathematicsENUMATH
- 2013

Numerical results for a driven cavity benchmark are presented, confirming that Parareal's convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection.

### A massively space-time parallel N-body solver

- Computer Science2012 International Conference for High Performance Computing, Networking, Storage and Analysis
- 2012

A novel space-time parallel version of the Barnes-Hut tree code PEPC is presented using PFASST, the Parallel Full Approximation Scheme in Space and Time, which relaxes the theoretical bound on parallel efficiency in parareal.

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