Integrating an N-Body Problem with SDC and PFASST

Speck, Robert; Ruprecht, Daniel; Krause, Rolf; Emmett, Matthew; Minion, Michael L.; Winkel, Mathias; Gibbon, Paul

doi:10.1007/978-3-319-05789-7_61

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…Thus, the integration of Boris-SDC into the novel PFASST++ framework [41] is planned for future work. This requires extending Boris-SDC to multiple levels in space and time with adequate coarsening strategies, see [25] for a description of multi-level SDC and [42,43] for a first idea for particlebased coarsening. An efficient time-parallel method tailored for particle simulations could greatly aid in better exploiting the computational resources of massively parallel high-performance computing systems for plasma physics applications.…”

Section: Discussionmentioning

confidence: 99%

A high-order Boris integrator

Winkel

Speck

Ruprecht

2015

Journal of Computational Physics

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This work introduces the high-order Boris-SDC method for integrating the equations of motion for electrically charged particles in an electric and magnetic field. Boris-SDC relies on a combination of the Boris-integrator with spectral deferred corrections (SDC). SDC can be considered as preconditioned Picard iteration to compute the stages of a collocation method. In this interpretation, inverting the preconditioner corresponds to a sweep with a low-order method. In Boris-SDC, the Boris method, a second-order Lorentz force integrator based on velocity-Verlet, is used as a sweeper/preconditioner. The presented method provides a generic way to extend the classical Boris integrator, which is widely used in essentially all particle-based plasma physics simulations involving magnetic fields, to a high-order method. Stability, convergence order and conservation properties of the method are demonstrated for different simulation setups. Boris-SDC reproduces the expected high order of convergence for a single particle and for the center-of-mass of a particle cloud in a Penning trap and shows good long-term energy stability.

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Section: Discussionmentioning

confidence: 99%

A high-order Boris integrator

Winkel

Speck

Ruprecht

2015

Journal of Computational Physics

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“…In these original papers from 2012 and 2014, the PFASST algorithm was introduced; its implementation was discussed and it was applied to first problems. In the following years, PFASST has been applied to more and more problems and coupled to different space‐parallel solver, ranging from a Barnes–Hut tree code to geometric multigrid . Together with spatial parallelization, PFASST was demonstrated to run and scale on up to 458,752 cores of an IBM Blue Gene/Q installation.…”

Section: Introductionmentioning

confidence: 99%

“…In the following years, PFASST has been applied to more and more problems and coupled to different space-parallel solver, ranging from a Barnes-Hut tree code to geometric multigrid. [12][13][14][15] Together with spatial parallelization, PFASST was demonstrated to run and scale on up to 458, 752 cores of an IBM Blue Gene/Q installation. Yet, while applications, implementation, and improvements are discussed frequently, a solid and reliable convergence theory is still missing.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

Bolten

Moser

Speck

2018

Numerical Linear Algebra App

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Summary For time‐dependent partial differential equations, parallel‐in‐time integration using the “parallel full approximation scheme in space and time” (PFASST) is a promising way to accelerate existing space‐parallel approaches beyond their scaling limits. Inspired by the classical Parareal method and multigrid ideas, PFASST allows to integrate multiple time steps simultaneously using a space–time hierarchy of spectral deferred correction sweeps. While many use cases and benchmarks exist, a solid and reliable mathematical foundation is still missing. Very recently, however, PFASST for linear problems has been identified as a multigrid method. In this paper, we will use this multigrid formulation and, in particular, PFASST's iteration matrix to show that, in the nonstiff and stiff limit, PFASST indeed is a convergent iterative method. We will provide upper bounds for the spectral radius of the iteration matrix and investigate how PFASST performs for increasing numbers of parallel time steps. Finally, we will demonstrate that the results obtained here indeed relate to actual PFASST runs.

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“…The FAS correction allows different strategies for spatial coarsening to be used within the mesh hierarchy, leading to improved parallel efficiency. An accuracy study of PFASST as well as serial SDC for a first-order particle-based discretization can be found in [8]. A successful demonstration of PFASST's efficacy in extreme-scale parallel particle simulations was presented in [9], where PFASST was combined with the parallel Barnes-Hut tree codes PEPC [10] to simulate a spherical vortex sheet.…”

Section: Introductionmentioning

confidence: 99%

A space-time parallel solver for the three-dimensional heat equation

Speck,

Ruprecht,

Emmett

et al. 2013

Preprint

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The paper presents a combination of the time-parallel "parallel full approximation scheme in space and time" (PFASST) with a parallel multigrid method (PMG) in space, resulting in a mesh-based solver for the three-dimensional heat equation with a uniquely high degree of efficient concurrency. Parallel scaling tests are reported on the Cray XE6 machine "Monte Rosa" on up to 16,384 cores and on the IBM Blue Gene/Q system "JUQUEEN" on up to 65,536 cores. The efficacy of the combined spatial-and temporal parallelization is shown by demonstrating that using PFASST in addition to PMG significantly extends the strong-scaling limit. Implications of using spatial coarsening strategies in PFASST's multi-level hierarchy in large-scale parallel simulations are discussed.

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Integrating an N-Body Problem with SDC and PFASST

Cited by 5 publications

References 23 publications

A high-order Boris integrator

A high-order Boris integrator

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

A space-time parallel solver for the three-dimensional heat equation

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