Convergence of Parareal for the Navier-Stokes Equations Depending on the Reynolds Number

Steiner, Johannes; Ruprecht, Daniel; Speck, Robert; Krause, Rolf

doi:10.1007/978-3-319-10705-9_19

Cited by 31 publications

(33 citation statements)

References 17 publications

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“…For these problems there is a high demand for highly efficient parallel solvers [53]. The Parareal algorithm was for example reported to perform poorly in advection-dominated problems [18,49]. Figure 4.6 illustrates that widening the spectrum does not significantly affect the convergence of the PEBK iterations.…”

Section: Parallel Efficiencymentioning

confidence: 99%

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Kooij

Botchev

Geurts

2017

Journal of Computational and Applied Mathematics

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Abstract. A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.Key words. parallel computing, exponential integrators, partial differential equations, parallel in time, block Krylov subspace.AMS subject classifications. 65F60, 65L05, 65Y051. Introduction. Recent developments in hardware architecture, bringing to practice computers with hundreds of thousands of cores, urge the creation of new, as well as a major revision of existing numerical algorithms [14]. To be efficient on such massively parallel platforms, the algorithms need to employ all possible means to parallelize the computations. When solving partial differential equations (PDEs) with a time-dependent solution, an important way to parallelize the computations is, next to the parallelization across space, parallelization across time. This adds a new dimension of parallelism with which the simulations can be implemented. In this paper we present a new time-parallel integration method extending the Paraexp method [21] to nonlinear partial differential equations using Krylov methods and waveform relaxation.Several approaches to parallelize the simulation of time-dependent solutions in time can be distinguished. The first important class of the methods are the waveform relaxation methods [6,38,44,56], including the space-time multigrid methods for parabolic PDEs [8,24,30,37]. The key idea is to start with an approximation to the numerical solution for the whole time interval of interest and update the solution, solving an easier-to-solve approximate system in time. The Parareal method [36], which attracted significant attention recently, is a prime example related to the class of waveform relaxation methods [19].Parallel Runge-Kutta methods and general linear methods, where the parallelism is determined and restricted by the number of stages or steps, form another class of the time-parallel methods [8,12,52]. Time-parallel schemes can also be obtained

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Section: Parallel Efficiencymentioning

confidence: 99%

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Kooij

Botchev

Geurts

2017

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Figure 4.16b indicates that a parallel speedup can be also achieved for ν = 10 −3 and ν = 10 −4 . The parallel efficiency of the Parareal algorithm is known to deteriorate for similarly small viscosity coefficients [161]. Furthermore, we note that the actual parallel efficiency of the PEBK method may even be better in practice, since the required number of PEBK iterations could decrease with higher P .…”

Section: Parallel Efficiencymentioning

confidence: 99%

“…Although, the performance in some cases can be improved by Krylov subspace enhancement [47,57,144]. In fluid flow problems, it was demonstrated that the performance of Parareal deteriorates in cases with low viscosity [49,100,161]. Direct numerical simulation at high Reynolds number then becomes challenging because convection in the turbulent flow becomes an important term and the high Reynolds number leads to small dissipation.…”

Section: Parallelism In Timementioning

confidence: 99%

“…For these problems there is a high demand for highly efficient parallel solvers [174]. The Parareal algorithm was for example reported to perform poorly in advection-dominated problems [49,161].…”

Section: Parallel Efficiencymentioning

confidence: 99%

“…A major disadvantage of Parareal is that the parallel efficiency deteriorates for hyperbolic problems [58,52]. Also for the Navier-Stokes equation at high Reynolds numbers strongly reduced performance was reported [161]. An interesting alternative, the Paraexp method, has been introduced in [54] for the time-parallel solution of linear initial-value problems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

Kooij¹

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Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection

Sterck

Falgout

Friedhoff

et al. 2021

Numerical Linear Algebra App

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Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge-Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping. K E Y W O R D Shigh-order, hyperbolic, MGRIT, multigrid, parallel-in-time, Parareal INTRODUCTIONAs compute clock speeds stagnate and core counts of parallel machines increase dramatically, parallel-in-time methods provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs). 1,2 Two well-known parallel time integration methods are Parareal 3 and multigrid reduction-in-time (MGRIT), 4

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Convergence of Parareal for the Navier-Stokes Equations Depending on the Reynolds Number

Cited by 31 publications

References 17 publications

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection

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