On the Stability and Accuracy of the Spectral Difference Method

Abeele, Kris Van Den; Lacor, Christian; Wang, Zhi Jian

doi:10.1007/s10915-008-9201-0

Cited by 145 publications

(104 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The result is consistent with the conclusion of Van den Abeele, Lacor and Wang [16] that the stability of the spectral difference method depends only on the location of the flux collocation points. While it establishes the stability of the SD scheme when the interior flux collocation points are the zeros of the Legendre polynomial L p (x), it does not preclude the stability of the SD scheme with other choices of the flux collocation points, possibly in a different norm.…”

Section: ) ∂X DXsupporting

confidence: 91%

“…While the SD method has proved robust and productive in a variety of applications [9][10][11][12][13][14][15], doubts have been raised about its stability. In particular it has been suggested that the SD scheme is not stable for higher order triangular elements [16], and sometimes weakly unstable in one dimension depending on the choice of flux collocation points.…”

mentioning

confidence: 99%

See 1 more Smart Citation

A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy

Jameson

2010

J Sci Comput

160

View full text Add to dashboard Cite

While second order methods for computational simulations of fluid flow provide the basis of widely used commercial software, there is a need for higher order methods for more accurate simulations of turbulent and vortex dominated flows. The discontinuous Galerkin (DG) method is the subject of much current research toward this goal. The spectral difference (SD) method has recently emerged as a promising alternative which can reduce the computational costs of higher order simulations. There remains some questions, however, about the stability of the SD method. This paper presents a proof that for the case of one dimensional linear advection the SD method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior fluxes collocation points are placed at the zeros of the corresponding Legendre polynomial.

show abstract

Section: ) ∂X DXsupporting

confidence: 91%

mentioning

confidence: 99%

A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy

Jameson

2010

J Sci Comput

160

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…Like CG and DG schemes (and for the same reasons), SD schemes are compact in nature. The intuitive nature of these methods, their apparent efficiency, and the presentation of formalized stability analysis by Van den Abeele, Lacor and Wang [129], and Jameson [63], has made SD schemes increasingly popular in recent years. For examples of their use see Liang, Jameson and Wang [79] and Ou et al [98].…”

Section: Spectral Difference 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.

View full text Add to dashboard Cite

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.

show abstract

On the Stability and Accuracy of the Spectral Difference Method

Cited by 145 publications

References 18 publications

A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy

A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy

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

Contact Info

Product

Resources

About