h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Klaij, C.M.; Raalte, M. H. van; Ven, H. van der; Vegt, J.J.W. van der

doi:10.1016/j.jcp.2007.08.034

Cited by 31 publications

(37 citation statements)

References 22 publications

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“…The coefficients b k ensure that the semi-implicit Runge-Kutta operator is the identity operator if v l nh;p is the exact steady state solution of (7). Without this condition the pseudo-time integration method would not converge to a steady state.…”

Section: Pseudo-time Runge-kutta Smoothermentioning

confidence: 99%

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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

Vegt

Rhebergen

2012

Journal of Computational Physics

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Section: Pseudo-time Runge-kutta Smoothermentioning

confidence: 99%

“…The remaining coefficients can be chosen such that the multigrid performance for a selected class of problems is optimal. In [7,12,15] we performed this optimization for explicit Runge-Kutta methods. In this section we will discuss the optimization of the semi-implicit Runge-Kutta smoother used in the hp-MGS multigrid algorithm.…”

Section: Multigrid Optimizationmentioning

confidence: 99%

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

Vegt

Rhebergen

2012

Journal of Computational Physics

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“…We do not discuss the details of the space-time DG discretization for the advection-diffusion equation, but refer to [3,5] for more details. In the multigrid optimization we consider a uniform space-time mesh with elements ∆t × ∆ x × ∆ y and periodic boundary conditions.…”

Section: Optimizing Multigrid For Space-time Dg Discretizationsmentioning

confidence: 99%

“…We start in Section 5.1 by comparing the optimized h-multigrid algorithms of the previous sections to the original EXI-EXV h-multigrid method [3]. For this we consider the 2D advection-diffusion equation.…”

Section: Testing Multigrid Performancementioning

confidence: 99%

“…In this paper we will discuss the optimization of multigrid techniques for higher order accurate space-time DG discretizations describing advection dominated flows. This research is a continuation of [3,7] where we presented a multigrid algorithm in combination with a pseudo-time integration method for second order accurate space-time DG discretizations of the compressible Euler and Navier-Stokes equations. The main benefits of this multigrid algorithm are that no large global linear system needs to be solved and, through the use of Runge-Kutta type smoothers, the locality of the DG discretization is preserved.…”

Section: Introductionmentioning

confidence: 99%

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Multigrid Optimization for Space-Time Discontinuous Galerkin Discretizations of Advection Dominated Flows

Rhebergen

Vegt

Ven

2010

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two-and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit RungeKutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p-and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated.

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Improved error and work estimates for high‐order elements

Löhner

2013

Numerical Methods in Fluids

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SUMMARY Work estimates for high‐order elements are derived. The comparison of error and work estimates shows that even for relative accuracy in the 0.1% range, which is one order below the typical accuracy of engineering interest (1% range), linear elements may outperform all higher‐order elements. As expected, the estimates also show that the optimal order of element in terms of work and storage demands depends on the desired relative accuracy. The comparison of work estimates for high‐order elements and their finite difference counterparts reveals a work‐ratio of several orders of magnitude. It thus becomes questionable if general geometric flexibility via micro‐unstructured grids is worth such a high cost. Copyright © 2013 John Wiley & Sons, Ltd.

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h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Cited by 31 publications

References 22 publications

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

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

Multigrid Optimization for Space-Time Discontinuous Galerkin Discretizations of Advection Dominated Flows

Improved error and work estimates for high‐order elements

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