# Space-time block preconditioning for incompressible flow

@article{Danieli2021SpacetimeBP, title={Space-time block preconditioning for incompressible flow}, author={Federico Danieli and Ben S. Southworth and Andrew J. Wathen}, journal={ArXiv}, year={2021}, volume={abs/2101.07003} }

Parallel-in-time methods have become increasingly popular in the simulation of timedependent numerical PDEs, allowing for the efficient use of additional MPI processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space…

## Figures and Tables from this paper

## 2 Citations

Parallel-in-time preconditioners for the Sinc-Nyström method

- Computer ScienceArXiv
- 2021

The proposed preconditioner is a low-rank perturbation of the original matrix and has two advantages: first, it is shown that the eigenvalues of the preconditionsed system are highly clustered with some uniform bounds which are independent of the mesh parameters, and second, it can be used parallel for all the Sinc time points via a block diagonalization procedure.

A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations

- Computer ScienceAdv. Comput. Math.
- 2022

A novel connection between the characteristic equation and the Chebyshev polynomials is presented, by which it is proved that Cond2(V ) = O(n2), which implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows and thus a much larger n can be used to yield satisfactory parallelism.

## References

SHOWING 1-10 OF 47 REFERENCES

Multigrid methods with space–time concurrency

- Computer ScienceComput. Vis. Sci.
- 2017

This paper develops parallel software and performance models to study the three basic options of multigrid algorithms on space–time grids that allow parallelism in space and time, and introduces an extension of one of them for handling multistep time integration.

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

- Computer Science
- 2006

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.

AIR algebraic multigrid for a space-time hybridizable discontinuous Galerkin discretization of advection(-diffusion)

- Computer ScienceSIAM J. Sci. Comput.
- 2021

A closer look at the geometric coarsening structure that arises in AIR explains why AIR can provide robust, scalable space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems.

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

- Computer ScienceHPSC
- 2012

This paper combines the Parareal parallel-in-time method together with spatial parallelization and investigates this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations, confirming that the space- time-parallel method can provide speedup beyond the saturation of the spatial parallelizations.

Parallel Time Integration with Multigrid

- Computer ScienceSIAM J. Sci. Comput.
- 2014

A nonintrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR) for solving diffusion equations in two and three space dimensions in numerical ex...

A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains

- Computer ScienceJ. Comput. Phys.
- 2012

A Direct Solver for Time Parallelization

- Computer Science
- 2016

This work presents here a mathematical analysis of a further method to parallelize in time, based on the diagonalization of the time stepping matrix, and proposes an optimization strategy for the choice of the parameters, for a Backward Euler discretized of the heat equation in two dimensions.

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

- Computer Science
- 2004

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.

A Space-Time Multigrid Method for Parabolic Partial Differential Equations

- Computer ScienceSIAM J. Sci. Comput.
- 1995

The solution of parabolic partial differential equations (PDEs) is considered, characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil.

50 Years of Time Parallel Time Integration

- Computer Science
- 2015

This chapter is for people who want to quickly gain an overview of the exciting and rapidly developing area of research of time parallel methods.