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Effect of space discretization on the parareal algorithm for advection-diffusion equations

MATHEMATICS AND COMPUTERS IN SIMULATION 2025年 233卷 330-340页

作者： Zeng, Xianfu Song, Haiyan Ningbo Univ Finance & Econ Coll Int Econ & Trade Ningbo 315175 Zhejiang Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Hubei Peoples R China NingboTech Univ Sch Comp & Data Engn Ningbo 315100 Zhejiang Peoples R China

The influence of time-integrator on the convergence rate of the parallel-in-time algorithm parareal has been extensively studied in literature, but the effect of space discretization was only rarely considered. In this paper, using the advection-diffusion equation parametrized by a diffusion coefficient nu > 0 as the model, we show that the space discretization indeed has a non-negligible effect on the convergence rate, especially when nu is small. In particular, for two space discretizations-the centered FD (finite difference) method and a Compact FD method of order 4, we show that the algorithm converges with very different rates, even though both the coarse and fine solvers of the algorithm are strongly stable under these two space discretizations. Numerical results for one-dimensional and two-dimensional cases are presented to validate the theoretical predictions.

关键词： parareal algorithm Convergence analysis Numerical stability Iterative methods Space discretization Advection-diffusion equation

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Multi-step variant of the parareal algorithm: convergence analysis and numerics

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS 2024年第2期58卷 673-694页

作者： Ait-Ameur, Katia Maday, Yvon Univ Montpellier CNRS Team LEMON IMAG Inria F-34095 Montpellier France Univ Paris Cite Sorbonne Univ CNRS Lab Jacques Louis Lions LJLL F-75005 Paris France Inst Univ France Paris France

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver's accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window.

关键词： Time domain decomposition multi-step time scheme parareal algorithm

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Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm

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JOURNAL OF COMPUTATIONAL PHYSICS 2010年第18期229卷 6558-6573页

作者： Samaddar, D. Newman, D. E. Sanchez, R. Univ Alaska Dept Phys Fairbanks AK 99775 USA Oak Ridge Natl Lab Div Fus Energy Oak Ridge TN 37831 USA

It is shown that numerical simulations of fully-developed plasma turbulence can be successfully parallelized in time using the parareal algorithm. The result is far from trivial, and even unexpected, since the exponential divergence of Lagrangian trajectories as well as the extreme sensitivity to initial conditions characteristic of turbulence set these type of simulations apart from the much simpler systems to which the parareal algorithm has been applied to this day. It is also shown that the parallel gain obtainable with this method is very promising (close to an order of magnitude for the cases and implementations described), even when it scales with the number of processors quite differently to what is typical for spatial parallelization. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Time parallelization parareal algorithm Turbulent transport Plasma turbulence Magnetically confined plasmas

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Application of the parareal algorithm to simulations of ELMs in ITER plasma

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COMPUTER PHYSICS COMMUNICATIONS 2019年 235卷 246-257页

作者： Samaddar, D. Coster, D. P. Bonnin, X. Berry, L. A. Elwasif, W. R. Batchelor, D. B. Culham Sci Ctr CCFE Abingdon OX14 3DB Oxon England Max Planck Inst Plasma Phys D-85748 Garching Germany ITER Org Route Vinon Verdon F-13115 St Paul Les Durance France Oak Ridge Natl Lab Oak Ridge TN USA

This paper explores the application of the parareal algorithm to simulations of ELMs in ITER plasma. The primary focus of this research is identifying the parameters that lead to optimum performance. Since the plasma dynamics vary extremely fast during an ELM cycle, a straightforward application of the algorithm is not possible and a modification to the standard parareal correction is implemented. The size of the time chunks also have an impact on the performance and needs to be optimized. A computational gain of 7.8 is obtained with 48 processors to illustrate that the parareal algorithm can be successfully applied to ELM plasma. Crown Copyright (C) 2018 Published by Elsevier B.V. All rights reserved.

关键词： Time parallelization parareal algorithm Scrape off layer (SOL) SOLPS Magnetically confined plasmas Edge localized modes or ELMs ITER Tokamak

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A parareal algorithm based on waveform relaxation

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MATHEMATICS AND COMPUTERS IN SIMULATION 2012年第11期82卷 2167-2181页

作者： Liu, Jun Jiang, Yao-Lin Xi An Jiao Tong Univ Dept Math Sci Xian 710049 Shaanxi Peoples R China China Univ Petr Coll Sci Qingdao 266580 Shandong Peoples R China

We report a new parallel iterative algorithm for time-dependent differential equations by combining the known waveform relaxation (WR) technique with the classical parareal algorithm. The parallelism can be simultaneously exploited in both sub-systems by WR and time by parareal. We also provide a sharp estimation on errors for the new algorithm. The iterations of parareal and WR are balanced to optimize the performance of the algorithm. Furthermore, the parallel speedup and efficiency of the new approach are analyzed by comparing with the classical parareal algorithm and the WR technique, respectively. Numerical experiments are carried out to verify the effectiveness of the theoretic work. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： parareal algorithm Waveform relaxation Parallelism in sub-systems and time Convergence analysis

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A DIAGONALIZATION-BASED parareal algorithm FOR DISSIPATIVE AND WAVE PROPAGATION PROBLEMS

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2020年第5期58卷 2981-3009页

作者： Gander, Martin J. WU, Shu-Lin Univ Geneva Sect Math CH-1211 Geneva Switzerland Northeast Normal Univ Sch Math & Stat Changchun 130024 Peoples R China

We present a new parareal algorithm based on a diagonalization technique proposed recently. The algorithm uses a single implicit Runge-Kutta method with the same small step-size for both the F and G propagators in parareal and would thus converge in one iteration when used directly like this, without, however, any speedup due to the sequential way parareal uses G. We then approximate G with a head-tail coupled condition such that G can be parallelized using diagonalization in time. We show that with a suitable choice of the parameter in the head-tail condition, our new parareal algorithm converges very rapidly, both for parabolic and hyperbolic problems, even in the nonlinear case. We illustrate our new algorithm with numerical experiments.

关键词： parareal algorithm diagonalization technique parallel coarse propagator dissipative problems wave propagation problems convergence analysis

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Scheduling of tasks in the parareal algorithm

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PARALLEL COMPUTING 2011年第3期37卷 172-182页

作者： Aubanel, Eric Univ New Brunswick Fac Comp Sci Fredericton NB E3B 5A3 Canada

Parallelization of partial differential equations (PDEs) by time decomposition has attracted much interest, mainly due to its potential to enable very long time simulations beyond what is possible using spatial domain decomposition. However, there has only been limited performance analysis of the parareal algorithm in the literature, ignoring the efficient scheduling of tasks. This paper presents a detailed study of the scheduling of tasks in the parareal algorithm that achieves significantly better efficiency than the usual algorithm. Two algorithms are proposed, one which uses a manager-worker paradigm with overlap of sequential and parallel phases, and a second that is completely distributed. Experiments were conducted with the 2D heat equation. It was found that the rate of convergence decreases as the number of processors increases, in the case of strong scaling (fixed time interval). However, for weak scaling results the rate of convergence was unaffected by the number of processors. The results of this paper suggest that the parareal algorithm is a promising approach to solving long time evolution problems, particularly when the goal is simulation of longer times using more processors. It also exhibits characteristics that make it particularly suitable for execution on heterogeneous computational grids, such as low communication overhead and easy accommodation of different processor and network speeds. (C) 2010 Elsevier B.V. All rights reserved.

关键词： parareal algorithm Time parallelization Numerical solution of PDEs

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CONVERGENCE ANALYSIS OF parareal algorithm BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2020年第3期38卷 487-501页

作者： Liying Zhang Jing Wang Weien Zhou Landong Liu Li Zhang School of Mathematical Science China University of Mining and TechnologyBeijing 100083China National Innovation Institute of Defense Technology Chinese Academy of Military ScienceBeijing 100101China Department of Foundation Courses Beijing Union UniversityBeijing 100101China

In this paper,we propose a parareal algorithm for stochastic differential equations(SDEs),which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine *** convergence order of the proposed algorithm is analyzed under some regular ***,numerical experiments are dedicated to illustrate the convergence and the convergence order with respect to the iteration number k,which show the efficiency of the proposed method.

关键词： Stochastic differential equations parareal algorithm Convergence Stochastic Taylor expansion Milstein scheme

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Analysis of the parareal algorithm for linear parametric differential equations

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2023年第4期100卷 775-795页

作者： Zhang, Ren-Hao Jiang, Yao-Lin Li, Jun Song, Bo Xi An Jiao Tong Univ Sch Math & Stat Xian Shaanxi Peoples R China Northwestern Polytech Univ Sch Math & Stat Xian Shaanxi Peoples R China Northwestern Polytech Univ Sch Math & Stat Xian 710072 Shaanxi Peoples R China

This paper presents a parareal algorithm with parameterized propagators for linear parametric differential equations over a wide range of parameters. Through transforming the initial value problem into nonparametric ODEs based on Taylor series, we construct the general parameterized fine and coarse propagators for the parareal algorithm in each time subinterval. Furthermore, to accelerate the convergence of the algorithm, the coarse propagator based on the waveform relaxation method is proposed. By analysing the computational complexity of the WR propagator and the general coarse propagator, we find these two propagators are appropriate for the different situations. Finally, the convergence analysis of the parareal algorithm with these propagators is presented and our analysis is illustrated with two numerical experiments.

关键词： parareal algorithm waveform relaxation method linear parametric differential equations convergence analysis numerical simulation

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

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COMPUTER PHYSICS COMMUNICATIONS 2017年 221卷 19-27页

作者： Samaddar, D. Coster, D. P. Bonnin, X. Bergmeister, C. Havlickova, E. Berry, L. A. Elwasif, W. R. Batchelor, D. B. Culham Sci Ctr CCFE Abingdon OX14 3DB Oxon England Max Planck Inst Plasma Phys Garching Germany Univ Paris 13 Sorbonne Paris Cite CNRS LSPM Villetaneuse France ITER Org Route Vinon Sur Verdon F-13115 St Paul Les Durance France Oak Ridge Natl Lab Oak Ridge TN USA

It is shown that numerical modelling of edge plasma physics may be successfully parallelized in time. The parareal algorithm has been employed for this purpose and the SOLPS code package coupling the B2.5 finite-volume fluid plasma solver with the kinetic Monte-Carlo neutral code Eirene has been used as a test bed. The complex dynamics of the plasma and neutrals in the scrape-off layer (SOL) region makes this a unique application. It is demonstrated that a significant computational gain (more than an order of magnitude) may be obtained with this technique. The use of the IPS framework for event-based parareal implementation optimizes resource utilization and has been shown to significantly contribute to the computational gain. (C) 2017 EURATOM/CCFE Fusion Association. Published by Elsevier B.V. All rights reserved.

关键词： Time parallelization parareal algorithm Edge plasma Scrape off layer (SOL) SOLPS Eirene Magnetically confined plasmas

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