Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere

Hamon, François P.; Schreiber, Martin; Minion, Michael L.

doi:10.1016/j.jcp.2018.09.042

Cited by 13 publications

(26 citation statements)

References 41 publications

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“…correction (SDC) [46,6,24], Parareal [34] and multigrid-reduction-in-time [12,7]. These methods have been developed for various application areas and with varying degree of intrusiveness, ease of implementation, level of parallelism, and potential for speedup.…”

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confidence: 99%

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

Hessenthaler¹,

Southworth²,

Nordsletten³

et al. 2020

SIAM J. Sci. Comput.

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This paper presents a multilevel convergence framework for multigrid-reduction-intime (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V-and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters.The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Key words. multilevel convergence theory, multigrid-reduction-in-time (MGRIT), parallel-intime, multigrid, analytic upper bounds, a priori estimates 1. Introduction. Modern computer architectures enable massively parallel computations for systems under numerical investigation. While clock rates of recent highperformance computing architectures have largely become stagnant, increased concurrency continues to reduce the time-to-solution, allowing for increased complexity of computational models and accuracy of computed quantities.Spatial domain decomposition (DD) methods are a wide-spread class of parallelization techniques to exploit parallelism in numerical simulations. Many DD methods are straightforward to implement and scalable in parallel up to the point that communication tasks become dominant over computation tasks. Thus, spatial parallelism may saturate without exploiting the full potential of the available hardware.Parallel-in-time methods [37,15] increase the amount of parallelism that can be exploited by introducing parallelism in the temporal domain. Many such methods exist, including waveform relaxation [35,47], space-time ...

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mentioning

confidence: 99%

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

Hessenthaler¹,

Southworth²,

Nordsletten³

et al. 2020

SIAM J. Sci. Comput.

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“…Next we compare the theoretical computational cost of PFASST-SH to those of the serial SDC and MLSDC-SH schemes (Hamon et al, 2019). The PFASST-SH integration scheme with a block of n ts time steps solved in parallel -which is assumed here to correspond to the number of processors -, M f + 1 temporal nodes on the fine level, M c + 1 temporal nodes on the coarse level, N PF iterations, and a spatial coarsening ratio between the number of coarse zonal wavenumbers and the number of fine zonal wavenumbers, α = R c /R f , is denoted by PFASST(n ts , M f + 1, M c + 1, N PF , α).…”

Section: Computational Cost Of Pfasst-shmentioning

confidence: 99%

“…This derivation assumes that the cost of computing the FAS correction at the end of Step B (a linear combination of function values already accounted for) is negligible. The computational cost of the serial SDC and MLSDC-SH are considered in Hamon et al (2019). Briefly, using the same block of n ts time steps of individual size ∆t PF = ∆t SDC the serial SDC(M f + 1, N S ) has a computational cost of…”

Section: Computational Cost Of Pfasst-shmentioning

confidence: 99%

“…Specifically, for an error norm of 3 × 10 −12 , PFASST(n ts = 16, M f + 1 = 5, M c + 1 = 3, N PF = 16, α = 1) can take the same time step size of 240 s as SDC(M f + 1 = 5, N S = 8), leading to a speedup of 3.7. For this error norm, MLSDC(M f + 1 = 5, M c + 1 = 3, N ML = 4, α = 4/5) is more expensive than the single-level SDC(M f + 1 = 5, N S = 8), as shown in Hamon et al (2019).…”

Section: Nonlinear Evolution Of An Unstable Barotropic Wavementioning

confidence: 99%

“…Specifically, we have shown in previous work that the SH basis can be truncated, or padded with zeros, to construct consistent spatial restriction and interpolation operators between levels. As in Hamon et al (2019), we exploit the properties of the SH transform to efficiently solve the implicit systems at each temporal node. This study is relevant for operational numerical weather prediction systems as the SH transform is used in the Integrated Forecast System (IFS) at the European Centre for Medium-Range Weather Forecasts (ECMWF, Wedi et al, 2013) and the Global Spectral Model (GSM) at the Japan Meteorological Agency (JMA, Kanamitsu et al, 1983).…”

Section: Introductionmentioning

confidence: 99%

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Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

Hamon

Schreiber

Minion

2020

Journal of Computational Physics

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The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field.We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel highorder fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space.The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes.

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Applications of time parallelization

Ong

Schroder

2020

Comput. Visual Sci.

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Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere

Cited by 13 publications

References 41 publications

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

Applications of time parallelization

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