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

Jacobus J.W. van der Vegt, Sander Rhebergen

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Abstract

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.
Original languageEnglish
Pages (from-to)7564-7583
Number of pages20
JournalJournal of computational physics
Volume231
Issue number22
DOIs
Publication statusPublished - 2012

Fingerprint

Discontinuous Galerkin
Advection
Runge-Kutta
advection
Mars Global Surveyor
Discretization
Multilevel Analysis
Higher Order
optimization
Optimization
Spectral Radius
operators
Advection-diffusion Equation
radii
Operator Norm
Semi-implicit
Finite Element Discretization
Thin Layer
Coefficient
coefficients

Keywords

  • Space–time methods
  • EWI-22712
  • Fourier analysis
  • Multi-level analysis
  • IR-83461
  • Discontinuous Galerkin methods
  • Higher order accurate discretizations
  • Runge–Kutta methods
  • METIS-293255
  • Multigrid algorithms

Cite this

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title = "HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part II: Optimization of the Runge-Kutta smoother",
abstract = "Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.",
keywords = "Space–time methods, EWI-22712, Fourier analysis, Multi-level analysis, IR-83461, Discontinuous Galerkin methods, Higher order accurate discretizations, Runge–Kutta methods, METIS-293255, Multigrid algorithms",
author = "{van der Vegt}, {Jacobus J.W.} and Sander Rhebergen",
note = "eemcs-eprint-22712",
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language = "English",
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journal = "Journal of computational physics",
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T1 - HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows

T2 - Part II: Optimization of the Runge-Kutta smoother

AU - van der Vegt, Jacobus J.W.

AU - Rhebergen, Sander

N1 - eemcs-eprint-22712

PY - 2012

Y1 - 2012

N2 - Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.

AB - Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.

KW - Space–time methods

KW - EWI-22712

KW - Fourier analysis

KW - Multi-level analysis

KW - IR-83461

KW - Discontinuous Galerkin methods

KW - Higher order accurate discretizations

KW - Runge–Kutta methods

KW - METIS-293255

KW - Multigrid algorithms

U2 - 10.1016/j.jcp.2012.05.038

DO - 10.1016/j.jcp.2012.05.038

M3 - Article

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SP - 7564

EP - 7583

JO - Journal of computational physics

JF - Journal of computational physics

SN - 0021-9991

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ER -