High-order discontinuous Galerkin methods using an hp-multigrid approach

Nastase, Cristian R.; Mavriplis, Dimitri J.

doi:10.1016/j.jcp.2005.08.022

Cited by 149 publications

(139 citation statements)

References 33 publications

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“…Studies to date indicate that using a judicious choice of explicit and implicit smoothers at various multigrid levels may help to circumvent memory related issues whilst maintaining a suitable rate of convergence [85] [80]. Also, studies indicate that a combination of both p-multigrid and geometric multigrid may be beneficial [91].…”

Section: Discussionmentioning

confidence: 99%

“…It was also found that such an approach lead to convergence rates independent of solution polynomial order. In 2006 Nastase and Mavriplis [91] developed a mixed geometric/p-multigrid approach that utilized various implicit smoothers. Specifically, when the zeroth order p-multigrid level was reached a geometric multigrid approach was employed to further coarsen the discretization.…”

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

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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“…The semi-coarsening multigrid is used as smoother in the h-MGS-multigrid in Algorithm 2. Using (10)- (11) and assuming that L P 2 we obtain the following estimate for the number of operations in the LU-decompositions in Algorithm 2 14 3 NMm…”

Section: Computational Complexitymentioning

confidence: 99%

“…During the past decade also extensive research into efficient multigrid algorithms for discontinuous Galerkin discretizations has been conducted, see e.g. [1,3,[8][9][10][11]. This research is motivated by the strong need in industrial applications to reduce the computational cost of DG methods.…”

Section: Introductionmentioning

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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“…h-and/or p-multigrid preconditioners are known to work well for diffusion-dominated problems, and they have been applied in a DG context [28,18,24]. However, in this work, the ILUT(p, τ ) is considered for the obtuse meshes, and its efficiency is compared with the ILU(0) results on the acute and obtuse meshes.…”

Section: Preconditioning On Obtuse Meshesmentioning

confidence: 99%

On the Impact of Triangle Shapes for Boundary Layer Problems Using High-Order Finite Element Discretization

Sun

Darmofal

Haimes

2010

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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The impact of triangle shapes, including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for high-order discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and high-order discretizations. Large angles are also found to be not problematic for the conditioning of the linear systems arising from the discretizations. Further, when choosing preconditioning strategies, coupling strengths among elements rather than element angle sizes should be taken into account. With an appropriate preconditioner, solutions on meshes with and without large angles can be achieved within a comparable time.

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High-order discontinuous Galerkin methods using an hp-multigrid approach

Cited by 149 publications

References 33 publications

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

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

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

On the Impact of Triangle Shapes for Boundary Layer Problems Using High-Order Finite Element Discretization

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