Geometric multigrid with applications to computational fluid dynamics

Wesseling, Pieter; Oosterlee, Cornelis W.

doi:10.1016/b978-0-444-50616-0.50013-0

Cited by 32 publications

(40 citation statements)

References 136 publications

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“…We also note that Trottenberg et al 30 suggested the specific augmentation of Vanka-style box relaxation in place of distributed relaxation near the domain boundaries. Comparing linear and bilinear interpolation, these results indicate that linear interpolation outperforms bilinear interpolation in this case, matching some existing studies 8,43,44 for other relaxation schemes. Table 4 shows that the measured multigrid convergence factors again match well with the LFA-predicted two-grid convergence factors for inexact Braess-Sarazin relaxation with Dirichlet boundary conditions and that the convergence is h-independent.…”

Section: Figure 11supporting

confidence: 89%

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

MacLachlan

2018

Numerical Linear Algebra App

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Summary Multigrid methods that use block‐structured relaxation schemes have been successfully applied to several saddle‐point problems, including those that arise from the discretization of the Stokes equations. In this paper, we present a local Fourier analysis of block‐structured relaxation schemes for the staggered finite‐difference discretization of the Stokes equations to analyze their convergence behavior. Three block‐structured relaxation schemes are considered: distributive relaxation, Braess–Sarazin relaxation, and Uzawa relaxation. In each case, we consider variants based on weighted‐Jacobi relaxation, as is most suitable for parallel implementation on modern architectures. From this analysis, optimal parameters are proposed, and we compare the efficiency of the presented algorithms with these parameters. Finally, some numerical experiments are presented to validate the two‐grid and multigrid convergence factors.

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Section: Figure 11supporting

confidence: 89%

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

MacLachlan

2018

Numerical Linear Algebra App

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“…In 1986, Vanka proposed a coupled multigrid method for the solution of the Navier-Stokes equations on structured, staggered grids [4]. The method used a coupled smoother on all grid levels that has come to be known as Symmetric Coupled Gauss-Seidel (SCGS) [5][6][7]. The method was shown to scale optimally (linearly) in terms of both work and storage for the problems investigated.…”

Section: Introductionmentioning

confidence: 99%

Multiple semi-coarsened multigrid method with application to large eddy simulation

Ham

Lien

Strong

2005

Int. J. Numer. Meth. Fluids

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SUMMARYThe Multiple Semi-coarsened Grid (MSG) multigrid method of Mulder (J. Comput. Phys. 1989; 83:303 -323) is developed as a solver for fully implicit discretizations of the time-dependent incompressible Navier-Stokes equations. The method is combined with the Symmetric Coupled Gauss-Seidel (SCGS) smoother of Vanka (Comput. Methods Appl. Mech. Eng. 1986; 55:321-338) and its robustness demonstrated by performing a number of large-eddy simulations, including bypass transition on a at plate and the turbulent thermally-driven cavity ow. The method is consistently able to reduce the non-linear residual by 5 orders of magnitude in 40-80 work units for problems with signiÿcant and varying coe cient anisotropy. Some discussion of the parallel implementation of the method is also included.

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“…Brandt 1984Brandt , 2001Trottenberg et al 2001;Thomas et al 2003). Present experience for steady problems indicates that good multigrid performance can nevertheless be obtained 2 , but at the expense of some complexity in the smoother, the spatial discretization scheme, and/or the coarse-grid correction algorithm (see the reviews of Wesseling & Oosterlee 2001;Thomas et al 2003).…”

Section: A47 Page 2 Of 30mentioning

confidence: 99%

“…If this were not the case, a decrease in the convergence speed due to inaccurate coarse-grid correction could result, as has been observed with finite volume discretizations in the original nonboundary-fitted (x, y) space (cf. Wesseling & Oosterlee 2001, and the references cited therein).…”

Section: Rationale For the Transformationmentioning

confidence: 99%

“…The FAS scheme derives its efficiency from the interplay between two crucial ingredients: a conventional iterative ("relaxation") scheme that acts as a smoother of high frequency components of the solution residual (in Fourier space), and a coarse-grid correction algorithm that annihilates the remaining low frequency modes of the residual, which are responsible for the slow convergence of conventional iterative methods (Brandt 1977(Brandt , 1984Wesseling & Oosterlee 2001;Trottenberg et al 2001). Both the smoother and the coarse-grid correction algorithm depend on the spatial discretization scheme with which one discretizes the equations.…”

Section: A47 Page 2 Of 30mentioning

confidence: 99%

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On multigrid solution of the implicit equations of hydrodynamics

Kifonidis

Müller

2012

A&A

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Aims. We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics. Schemes of this class are free of the Courant-Friedrichs-Lewy condition. They are intended for simulations in which widely differing wave propagation timescales are present. A preferred solver in this class is identified. Applications to some simple stiff test problems that are governed by the compressible Euler equations, are presented to evaluate the convergence behavior, and the stability properties of this solver. Algorithmic areas are determined where further work is required to make the method sufficiently efficient and robust for future application to difficult astrophysical flow problems. Methods. The basic equations are formulated and discretized on non-orthogonal, structured curvilinear meshes. Roe's approximate Riemann solver and a second-order accurate reconstruction scheme are used for spatial discretization. Implicit Runge-Kutta (ESDIRK) schemes are employed for temporal discretization. The resulting discrete equations are solved with a full-coarsening, nonlinear multigrid method. Smoothing is performed with multistage-implicit smoothers. These are applied here to the time-dependent equations by means of dual time stepping. Results. For steady-state problems, our results show that the efficiency of the present approach is comparable to the best implicit solvers for conservative discretizations of the compressible Euler equations that can be found in the literature. The use of red-black as opposed to symmetric Gauss-Seidel iteration in the multistage-smoother is found to have only a minor impact on multigrid convergence. This should enable scalable parallelization without having to seriously compromise the method's algorithmic efficiency. For time-dependent test problems, our results reveal that the multigrid convergence rate degrades with increasing Courant numbers (i.e. time step sizes). Beyond a Courant number of nine thousand, even complete multigrid breakdown is observed. Local Fourier analysis indicates that the degradation of the convergence rate is associated with the coarse-grid correction algorithm. An implicit scheme for the Euler equations that makes use of the present method was, nevertheless, able to outperform a standard explicit scheme on a time-dependent problem with a Courant number of order 1000. Conclusions. For steady-state problems, the described approach enables the construction of parallelizable, efficient, and robust implicit hydrodynamics solvers. The applicability of the method to time-dependent problems is presently restricted to cases with moderately high Courant numbers. This is due to an insufficient coarse-grid correction of the employed multigrid algorithm for large time steps. Further research will be required to help us to understand and overcome the observed multigrid convergence difficulties for time-dependent problems.

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Geometric multigrid with applications to computational fluid dynamics

Cited by 32 publications

References 136 publications

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

Multiple semi-coarsened multigrid method with application to large eddy simulation

On multigrid solution of the implicit equations of hydrodynamics

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