Mechanisms for the convergence of time-parallelized, parareal turbulent plasma simulations

Reynolds-Barredo, J. M.; Newman, D. E.; Sánchez, Raúl; Samaddar, Debasmita; Berry, L. A.; Elwasif, Wael

doi:10.1016/j.jcp.2012.07.028

Cited by 36 publications

(33 citation statements)

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“…Iterations are then performed over the entire solution history, aiming to converge to the solution of the initial value problem. In particular, the Parareal method has been demonstrated to converge for large scale turbulent plasma simulations 9,10 . However, many time parallel methods suffer from a common scalability barrier in the presence of chaotic dynamics.…”

Section: Fig 3: Illustration Of Spatial Parallelism (Left) and Spacementioning

confidence: 99%

“…It also has a energy minimization form, allowing a standard finite element discretization. Appendix B details the discretization and the numerics of solving Equation (9). We implemented both a direct solver and an iterative solver for the symmetric boundary value problem (9).…”

Section: Update To Transformed Time By Letting τ = T; Continue Tomentioning

confidence: 99%

“…Our naive implementation of multigridin-time requires an n iter ranging from 100 to 1000 for the Lorenz system and the Kuramoto-Sivashinsky equation. This number could be reduced through more research into iterative solution algorithm specifically for Equation (9). When the operator L is a large scale spatial operator, we envision that the multigrid-in-time method can be extended to a space-time multigrid method.…”

Section: Update To Transformed Time By Letting τ = T; Continue Tomentioning

confidence: 99%

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Towards scalable parallel-in-time turbulent flow simulations

et al. 2013

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We present a reformulation of unsteady turbulent flow simulations. The initial condition is relaxed and information is allowed to propagate both forward and backward in time. Simulations of chaotic dynamical systems with this reformulation can be proven to be well-conditioned time domain boundary value problems. The reformulation can enable scalable parallel-in-time simulation of turbulent flows. I. NEED FOR SPACE-TIME PARALLELISMThe use of computational fluid dynamics (CFD) in science and engineering can be categorized into Analysis and Design. A CFD Analysis performs a simulation on a set of manually picked parameter values. The flow field is then inspected to gain understanding of the flow physics. Scientific and engineering decisions are then made based on understanding of the flow field. Analysis based on high fidelity turbulent flow simulations, particular Large Eddy Simulations, is a rapidly growing practice in complex engineering applications 12 .CFD based Design goes beyond just performing individual simulations, towards sensitivity analysis, optimization, control, uncertainty quantification and data based inference. Design is enabled by Analysis capabilities, but often requires more rapid turnaround. For example, an engineer designer or an optimization software needs to perform a series of simulations, modifying the geometry based on previous simulation results. Each simulation must complete within at most a few hours in an industrial design environment. Most current practices of design use steady state CFD solvers, employing RANS (Reynolds Averaged NavierStokes) models for turbulent flows. Design using high fidelity, unsteady turbulent flow simulations has been investigated in academia 3 . Despite their great potential, high fidelity design is infeasible in an industrial setting because each simulation typically takes days to weeks. FIG. 1:Exponential increase of high performance computing power, primarily sustained by increased parallelism in the past decade. Data originate from top500.org. GFLOPS, TFLOPS, PFLOPS and EFLOPS represent 10 9 , 10 12 , 10 15 and 10 18 FLoating point Operations Per Second, respectively.The inability of performing high fidelity turbulent flow simulations in short turnaround time is a barrier to the game-changing technology of high fidelity CFD-based design. Nevertheless, development in High Performance Computing (HPC), as shown in Figure 1, promises to delivery in about ten years computing hardware a thousand times faster than those available today. This will be achieved through extreme scale a) Corresponding

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Section: Fig 3: Illustration Of Spatial Parallelism (Left) and Spacementioning

confidence: 99%

Section: Update To Transformed Time By Letting τ = T; Continue Tomentioning

confidence: 99%

Section: Update To Transformed Time By Letting τ = T; Continue Tomentioning

confidence: 99%

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Towards scalable parallel-in-time turbulent flow simulations

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“…Moreover, it must be clarified that temporal parallelization is not a replacement for any existing form of parallelization, but may be added to them to further, often significantly, improve performance. Temporal parallelization using the parareal algorithm was originally proposed in [1] in 2001, and has been applied to a variety of problems from relatively simple ones [2,3,4,5,6,7,8] to ones with complex dynamics [9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

“…Fully developed turbulence simulations in the steady state where the statistical study of properties such as the probability distribution function, vorticity and velocity are relevant are dealt with in [9,10]. [11] applies to burning plasma scenarios but the details of the physics of the edge involving impurities and neutral transport are not given significant importance.…”

Section: Introductionmentioning

confidence: 99%

Temporal parallelization of edge plasma simulations using the parareal algorithm and the SOLPS code

Samaddar

Coster

Bonnin

et al. 2017

Computer Physics Communications

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Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

2017

Math Methods in App Sciences

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Time‐dependent PDEs with fractional Laplacian ( − Δ)α play a fundamental role in many fields and approximating ( − Δ)α usually leads to ODEs' system like u′(t) + Au(t) = g(t) with A = Qα, where Q∈double-struckRm×m is a sparse symmetric positive definite matrix and α > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators scriptG and scriptF. The propagators scriptG and scriptF are respectively associated with large step size ΔT and small step size Δt, where ΔT = JΔt and J⩾2 is an integer. If we fix the scriptG‐propagator to the Implicit‐Euler method and choose for scriptF some proper Runge–Kutta (RK) methods, such as the second‐order and third‐order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy ρ≈13,∀J⩾2 and ∀σ(A)⊂[0,+∞), where σ(A) is the spectrum of the matrix A. In this paper, we show that by choosing these two RK methods as the scriptF‐propagator, the convergence factors can reach 112, provided the one‐stage complex Rosenbrock method is used as the scriptG‐propagator. If we choose for both scriptG and scriptF, the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach 112. Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

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Mechanisms for the convergence of time-parallelized, parareal turbulent plasma simulations

Cited by 36 publications

References 14 publications

Towards scalable parallel-in-time turbulent flow simulations

Towards scalable parallel-in-time turbulent flow simulations

Temporal parallelization of edge plasma simulations using the parareal algorithm and the SOLPS code

Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

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