Multigrid method based on a space-time approach with standard coarsening for parabolic problems

Franco, Sebastião Romero; Gaspar, Francisco José; Pinto, Márcio Augusto Villela; Rodrigo, Carmen

doi:10.1016/j.amc.2017.08.043

Cited by 16 publications

(12 citation statements)

References 16 publications

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“…The Multigrid method is a technique commonly adopted by a number of researchers when solving Au = f equation systems, which regularly require high computational cost, Fedorenko [17] [18], Franco et al [19], Wesseling [20], Pinto et al…”

Section: Multigrid Methodsmentioning

confidence: 99%

Computational simulation of one-dimensional waves with the Multigrid Method / Simulação computacional de ondas unidimensionais com o Método Multigrid

Malacarne¹,

Pinto²,

Franco³

2021

BJDV

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Several Engineering problems are modeled computationally, these simulations involve large systems, which are commonly difficult to solve. This paper deals with the simulation of one-dimensional waves, where the system resulting from the discretization by the Finite Difference Method is solved using the Multigrid Method with the conventional Gauss-Seidel solver, in order to decrease the computational time. Temporal discretization using the Time-Stepping method, where the system of equations is solved at each time step sequentially.

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Section: Multigrid Methodsmentioning

confidence: 99%

Computational simulation of one-dimensional waves with the Multigrid Method / Simulação computacional de ondas unidimensionais com o Método Multigrid

Malacarne¹,

Pinto²,

Franco³

2021

BJDV

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“…Space-time multigrid to solve time dependent partial differential equations (PDEs) traces back to [10,15], and has been further investigated in, e.g., [6,9,11,12,13,19]; see also [5] for a nice overview. With this approach, instead of considering one time step at a time, one forms an extended system with the whole set of discrete variables, to which a multigrid method is applied.…”

Section: Introductionmentioning

confidence: 99%

“…Our analysis is, to our knowledge, more rigorous and thorough than previous works about coarsening in space. In [6,8,12,19], relatively accurate convergence estimates are provided, but they are based on local Fourier analysis (LFA), and, therefore, heuristic for parabolic problems, as this approach is by nature unable to track rigorously nonnormality effects. Moreover, because of the many parameters involved, analytical studies are intractable.…”

Section: Introductionmentioning

confidence: 99%

Rigorous convergence proof of space-time multigrid with coarsening in space

Notay

2021

Numer Algor

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Space-time multigrid refers to the use of multigrid methods to solve discretized partial differential equations considering at once multiple time steps. A new theoretical analysis is developed for the case where one uses coarsening in space only. It proves bounds on the 2-norm of the iteration matrix that connect it to the norm of the iteration matrix when using the same multigrid method to solve the corresponding stationary problem. When using properly defined wavefront type smoothers, the bound is uniform with respect to the mesh size, the time step size, and the number of time steps, and addresses both the two-grid case and the W-cycle. On the other hand, for time-parallel smoothers, the results clearly show the condition to be satisfied by the time step size to have similar performance as with wavefront type smoothers. The analysis also leads to the definition of an effective smoothing factor that allows one to quickly check the potentialities of a given smoothing scheme. The accuracy of the theoretical estimates is illustrated on a numerical example, highlighting the relevance of the effective smoothing factor and the usefulness in following the provided guidelines to have robustness with respect to the time step size.

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“…The objective of this work is to compare two of the most relevant recent approaches: PFASST [14] and space-time multigrid methods (STMG) [26,29,18,24,21,6]. The current paper is similar in spirit to the comparison presented in [19] (where the authors suggest a future comparison with PFASST).…”

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confidence: 99%

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

Benedusi¹,

Minion²,

Krause³

2020

Preprint

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We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multilevel strategies. For both approaches, we start from an integral formulation of the continuous time dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multigrid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies is compared for varying order of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences.

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Multigrid method based on a space-time approach with standard coarsening for parabolic problems

Cited by 16 publications

References 16 publications

Computational simulation of one-dimensional waves with the Multigrid Method / Simulação computacional de ondas unidimensionais com o Método Multigrid

Computational simulation of one-dimensional waves with the Multigrid Method / Simulação computacional de ondas unidimensionais com o Método Multigrid

Rigorous convergence proof of space-time multigrid with coarsening in space

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

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