HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

Vegt, J.J.W. van der; Rhebergen, Sander

doi:10.1016/j.jcp.2012.05.037

Cited by 20 publications

(15 citation statements)

References 16 publications

(22 reference statements)

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“…This analysis provides the operator norm and spectral radius of the error transformation operator of the hp-MGS algorithm. This information is important both to obtain realistic estimates of the multigrid performance and to optimize the multigrid algorithm, which will be discussed in Part II [32]. The analysis of the hp-MGS algorithm is demonstrated for algebraic systems resulting from a fourth order accurate space-time DG discretization of the two-dimensional advection-diffusion equation for various cell Reynolds numbers and mesh aspect ratios.…”

Section: Discussionmentioning

confidence: 99%

“…The multilevel analysis shows that the new hp-MGS algorithm has excellent convergence rates for a wide range of cell Reynolds numbers, both on uniform and stretched meshes and for steady and time-dependent problems. The hp-MGS(1) and the standard hp-multigrid algorithm do not perform well for high cell Reynolds numbers despite extensive optimization conducted in Part II [32]. At low cell Reynolds numbers, the hp-MGS, hp-MGS(1) and the hp-multigrid algorithms converge well.…”

Section: Discussionmentioning

confidence: 99%

“…The optimization of the Runge-Kutta smoother, including details about the Runge-Kutta coefficients, and the performance of the hp-MGS algorithm on a number of test cases is discussed in Part II [32].…”

Section: Discrete Fourier Transform Of Hp-mgs Multigrid Error Transfomentioning

confidence: 99%

“…The need for improved convergence rates in the iterative solution of higher order accurate DG discretizations has motivated the research presented in this and the companion article [32], to which we will refer as Part II. The objectives of this research are to develop, analyze and optimize new multigrid algorithms for higher order accurate space-time DG discretizations of advection dominated flows.…”

Section: Introductionmentioning

confidence: 99%

“…The multigrid performance of the hp-MGS(1) and the hp-multigrid algorithm is very poor and the algorithms are not suitable for higher order accurate DG discretizations, despite extensive optimization of the semi-implicit Runge-Kutta smoother. A more detailed comparison of the different multigrid algorithms and their computational cost will be given in Part II [32].…”

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confidence: 99%

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hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

Vegt¹,

Rhebergen²

2012

Journal of Computational Physics

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a b s t r a c tThe hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-time discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time discontinuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for steady state and time-dependent problems, and low and high cell Reynolds numbers, including highly stretched meshes.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discrete Fourier Transform Of Hp-mgs Multigrid Error Transfomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

Vegt¹,

Rhebergen²

2012

Journal of Computational Physics

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Efficient p‐multigrid discontinuous Galerkin solver for complex viscous flows on stretched grids

Ghidoni

Colombo

Bassi

et al. 2014

Numerical Methods in Fluids

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SUMMARYDiscontinuous Galerkin methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids but are rather demanding in terms of computational resources. In order to improve their computational efficiency, a p‐multigrid solution strategy is here considered for the solution of the Navier–Stokes equations. In particular, a line smoother will be used to alleviate the effect of stretched grids on the convergence rate. The effectiveness and efficiency of the proposed approach in the solution of compressible shockless flow problems is demonstrated on the following 3D viscous test cases: a Delta Wing and a 3D streamlined body. Copyright © 2014 John Wiley & Sons, Ltd.

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A locally conservative and energy‐stable finite‐element method for the Navier‐Stokes problem on time‐dependent domains

Horváth

Rhebergen

2019

Numerical Methods in Fluids

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Summary We present a finite‐element method for the incompressible Navier‐Stokes problem that is locally conservative, energy‐stable, and pressure‐robust on time‐dependent domains. To achieve this, the space‐time formulation of the Navier‐Stokes problem is considered. The space‐time domain is partitioned into space‐time slabs, which in turn are partitioned into space‐time simplices. A combined discontinuous Galerkin method across space‐time slabs and space‐time hybridized discontinuous Galerkin method within a space‐time slab results in an approximate velocity field that is H(div)‐conforming and exactly divergence‐free, even on time‐dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.

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HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

Cited by 20 publications

References 16 publications

hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

Efficient p‐multigrid discontinuous Galerkin solver for complex viscous flows on stretched grids

A locally conservative and energy‐stable finite‐element method for the Navier‐Stokes problem on time‐dependent domains

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