A Reconstruction Approach to High-Order Schemnes Including Discontinuous Galerkin for Diffusion

Huynh, H. T.

doi:10.2514/6.2009-403

Cited by 196 publications

(186 citation statements)

References 18 publications

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“…Several additional forms of g L (and thus g R ) have also been suggested, leading to the development of new schemes, with various stability and accuracy properties. For further details of these new schemes see the articles by Huynh [1,2].…”

Section: Overview Of the Flux Reconstruction Approachmentioning

confidence: 99%

“…Utilizing the FR approach of Huynh [1,2], it was proved by Jameson [6] that (for 1D linear advection) a particular SD method is stable for all orders of accuracy in a broken norm of Sobolev type. Recently, this result has been extended by Vincent, Castonguay and Jameson [7], who identified a class of FR schemes which are guaranteed to be linearly stable.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Overview Of the Flux Reconstruction Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Non-linear Stability of Flux Reconstruction Schemes

2011

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“…The most attractive feature of rDG is the simplicity and generality of the recovery principle, which immediately allows one to create and further develop DG diffusion schemes. These schemes are of higher accuracy than any other DG diffusion schemes currently in use, while boasting the smallest eigenvalues, hence the largest stability range for explicit time-marching schemes, as shown by Huynh 34 . However, one significant weakness of rDG methods is a lack of flexibility.…”

Section: Introductionmentioning

confidence: 99%

A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

Luo

Nourgaliev

et al. 2010

Journal of Computational Physics

159

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A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi-Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier-Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier-Stokes equations.

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“…The FR approach potentially offers an equally simple way to treat viscous terms [59]. The present authors and coworkers are currently investigating whether a simple FR based Navier-Stokes flow solver can be developed for 2D and 3D unstructured meshes.…”

Section: Flux Reconstruction For Viscous Termsmentioning

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 Reconstruction Approach to High-Order Schemnes Including Discontinuous Galerkin for Diffusion

Cited by 196 publications

References 18 publications

On the Non-linear Stability of Flux Reconstruction Schemes

On the Non-linear Stability of Flux Reconstruction Schemes

A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

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

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