Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection–diffusion equation

Mascarenhas, Brendan S.; Helenbrook, Brian T.; Atkins, Harold L.

doi:10.1016/j.jcp.2010.01.020

Cited by 15 publications

(13 citation statements)

References 22 publications

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“…Applications of p-multigrid to higher order accurate DG discretizations of advection dominated flows can be found in [2,8,[17][18][19]21]. The resulting algebraic system at the coarsest p-multigrid level can, however, still be very large.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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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“…Therefore, this approach has been suggested as a solver algorithm for inviscid and laminar viscous flows by many authors. [12][13][14][15][16] However, our findings indicate a that in the context of turbulent flows the nonlinear p-multigrid is not a stable algorithm. 8 In this paper a nonlinear p-multigrid will be investigated further with additional alteration, e. g. using a Galerkin-transfer for the Jacobian on the lower levels.…”

Section: Introductionmentioning

confidence: 71%

Higher Order Multigrid Algorithms for a Discontinuous Galerkin RANS Solver

Wallraff

Leicht

2014

52nd Aerospace Sciences Meeting

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A comparison of nonlinear and linear p-and h-multigrid algorithms will be presented with respect to both algorithmic convergence properties and run-time behavior. In addition to that a comparison with a single-level solver, namely a Backward-Euler method, will be presented as well. The algorithms will be used to solve the RANS-equations in combination with a k-ω turbulence models for CFD applications in the area of compressible aerodynamic flows with high Reynolds numbers. The h-multigrid algorithms are formulated on agglomerated unstructured meshes. Results will be presented on both structured and unstructured meshes.

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“…It was found that the rate of convergence could be increased by a factor of 5-10 when using an implicit smoother (relative to using an explicit RK smoother). Also in 2010, Mascarenhas, Helenbrook and Atkins [87] extended their previous study [49] to investigate the coupling of p-multigrid and geometric multigrid in a DG context when solving the convection diffusion equation. Once again they focused particular attention on the performance of the first to zeroth order p-multigrid step.…”

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Vincent

Jameson

2011

Math. Model. Nat. Phenom.

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Abstract. Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of high-order methods amongst a non-specialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat towards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

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Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection–diffusion equation

Cited by 15 publications

References 22 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

Higher Order Multigrid Algorithms for a Discontinuous Galerkin RANS Solver

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

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