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

  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},
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… 

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

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

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

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

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

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

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

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

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

  • R. SpeckD. Ruprecht P. Gibbon
  • Computer Science
    2012 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.



Parallel-in-Time Simulation of Two-Dimensional, Unsteady, Incompressible Laminar Flows

An application of a parallel-in-time algorithm to the solution of the unsteady incompressible Navier-Stokes equations and an extension of the algorithm to parallelize simultaneously the space and time domains is presented.

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

A parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type is presented and significant computer time saving was achieved when compared with the single processor computing time.

Analysis of the Parareal Time-Parallel Time-Integration Method

New convergence results that show superlinear convergence of the parareal algorithm when used on bounded time intervals, and linear convergence for unbounded intervals are shown.

Numerical solution of the Navier-Stokes equations

A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the

A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations

This paper adapts the “parareal in time” algorithm to solve the challenging Navier-Stokes problem, with a coarse solver, based on a larger timestep, that helps to preserve stability and provides for more significant savings.

Time‐decomposed parallel time‐integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications

This methodology parallelizes the time-loop of a time- dependent partial differential equation solver without interfering with its sequential or parallel space-computations for time-dependent problems with a few degrees of freedom such as those arising in robotics and protein folding applications.


This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within theParareal iterations that utilizes the parallel speedup and efficiency of the hybrid methods.

Curvature‐compensated convective transport: SMART, A new boundedness‐ preserving transport algorithm

A polynomial-based discretization scheme is constructed around a technique called ‘curvature compensation’; the resultant curvature-compensated convective transport approximation is essentially third-order accurate in regions of the solution domain where the concept of order is meaningful.

Domain Decomposition Methods in Science and Engineering

Invited Talks.- Non-matching Grids and Lagrange Multipliers.- A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems.- Hybrid Schwarz-Multigrid Methods for the Spectral

Numerical Simulation in Fluid Dynamics: A Practical Introduction

Numerical simulation - a key technology of the future and example applications for free boundary value problems, and the mathematical description of flows.