A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows

Kopriva, David A.; Kolias, John H.

doi:10.1006/jcph.1996.0091

Cited by 438 publications

(269 citation statements)

References 34 publications

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“…They offer another route which avoids the need for calculating integrals and therefore defining quadrature rules, making these schemes attractive from an implementation point of view. The first of these, namely the spectral-difference scheme, was introduced by Kopriva and Kolias [5] in 1996 and was extended to quadrilateral and triangular elements by Liu et al [6] in 2006. Most recently, the flux reconstruction (FR) method was presented by Huynh [7].…”

Section: Introductionmentioning

confidence: 99%

On the Connections Between Discontinuous Galerkin and Flux Reconstruction Schemes: Extension to Curvilinear Meshes

et al. 2015

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This paper investigates the connections between many popular variants of the well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-product meshes. We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over-or consistent-integration-based dealiasing methods. The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over-or consistent-integration in an equivalent manner for both the approaches.

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Section: Introductionmentioning

confidence: 99%

On the Connections Between Discontinuous Galerkin and Flux Reconstruction Schemes: Extension to Curvilinear Meshes

et al. 2015

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“…Recently Huynh [1,2] proposed the flux reconstruction (FR) approach, which encompasses both collocation based nodal DG schemes of the type described by Hesthaven and Warburton [3], and spectral difference (SD) methods (at least for a linear flux function), which were originally proposed by Kopriva and Kolias [4], and later generalized by Liu, Vinokur and Wang [5].…”

Section: Introductionmentioning

confidence: 99%

On the Non-linear Stability of Flux Reconstruction Schemes

2011

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The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG) methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently a new range of linearly stable FR schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. In this short note non-linear stability properties of FR schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability). It is shown that linearly stable VCJH schemes (at least in their standard form) may be unstable if the flux function is non-linear. This instability is due to aliasing errors, which manifest since FR schemes (in their standard form) utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. Strategies for minimizing such aliasing driven instabilities are discussed within the context of the FR approach. In particular, it is shown that the location of the solution points will have a significant effect on non-linear stability. This result is important, since linear analysis of FR schemes implies stability is independent of solution point location. Finally, it is shown that if an exact L2 projection is employed to construct an approximation of the flux (as opposed to a collocation projection), then aliasing errors and hence aliasing driven instabilities will be eliminated. However, performing such a projection exactly, or at least very accurately, would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the FR approach. It can be noted that in all above regards, non-linear stability properties of FR schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of FR schemes, which have hitherto been developed and analyzed solely in the context of a linear flux function.

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“…The foundation for SD schemes was first put forward by Kopriva and Kolias [76] in 1996 under the name of "staggered grid Chebyshev multidomain" methods. However, several years later in 2006 Liu, Vinokur and Wang [83] presented a more general formulation for both triangular and quadrilateral elements, which they termed the SD method (a name which has been retained to the present).…”

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.

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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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A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows

Cited by 438 publications

References 34 publications

On the Connections Between Discontinuous Galerkin and Flux Reconstruction Schemes: Extension to Curvilinear Meshes

On the Connections Between Discontinuous Galerkin and Flux Reconstruction Schemes: Extension to Curvilinear Meshes

On the Non-linear Stability of Flux Reconstruction Schemes

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

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