Dispersion and dissipation properties of the 1D spectral volume method and application to a p-multigrid algorithm

Abeele, Kris Van Den; Broeckhoven, Tim; Lacor, Christian

doi:10.1016/j.jcp.2006.10.022

Cited by 67 publications

(43 citation statements)

References 19 publications

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“…The application of eigensolution analysis to spectral element methods is not new [24,[34][35][36][37][38][39][40] . Nevertheless, most of the dedicated literature covers only the temporal eigenanalysis approach, which (strictly speaking) is valid more effectively for periodic problems.…”

Section: Introductionmentioning

confidence: 99%

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

et al. 2018

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a b s t r a c tThe study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods -specifically discontinuous Galerkin (DG) -via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence.

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Section: Introductionmentioning

confidence: 99%

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

et al. 2018

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“…Once again order-independent convergence rates were observed. In 2007 Van den Abeele, Broeckhoven and Lacor [127] applied a p-multigrid algorithm with an explicit RK smoother in a SV context in 1D. Also in 2007 Wolkov, Hirsch and Leonard [144] applied a p-multigrid algorithm in a DG context to solve the 3D Euler and Navier-Stokes equations on unstructured hexahedral grids.…”

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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“…Most of this work is either with spectral element methods or with the discontinuous Galerkin method for fluid flow problems. At this point it becomes difficult to be exhaustive in the citations, but a sampling of the work can be found in [26][27][28][29][30][31][32][33][34][35][36][37].…”

Section: Historymentioning

confidence: 99%

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

Mitchell

2010

Numerical Linear Algebra App

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SUMMARYRecently the hp version of the finite element method, in which adaptivity occurs in both the size, h, of the elements and in the order, p, of the approximating piecewise polynomials, has received increasing attention. It is desirable to combine this optimal order discretization method with an optimal order algebraic solution method like multigrid. An intriguing notion is to use the values of p as the levels of a multilevel method. In this paper we present such a method, known as hp-multigrid, for high-order finite elements and hp-adaptive grids. We present a survey of the development of p-multigrid and hp-multigrid, define an hp-multigrid algorithm based on the p-hierarchical basis for the p levels and h-hierarchical basis for an h-multigrid solution of the p = 1 'coarse grid' equations, and present numerical convergence results using hp-adaptive grids. The numerical results suggest that the method has a convergence rate of

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Dispersion and dissipation properties of the 1D spectral volume method and application to a p-multigrid algorithm

Cited by 67 publications

References 19 publications

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

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

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

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