A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements

Castonguay, Patrice; Vincent, Peter; Jameson, Antony

doi:10.1007/s10915-011-9505-3

Cited by 141 publications

(135 citation statements)

References 24 publications

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“…There are several different FR approaches for solving first-order systems on triangles (see Huynh,45 Castonguay et al, 36 and Gao et al 40 ). To the author's knowledge, the approach of Castonguay et al is the only extension of FR that is provably stable for all orders of accuracy (in the context of linear first-order systems on triangles).…”

Section: B Extension Of Fr Approach To Trianglesmentioning

confidence: 99%

“…(Full details of the vector corrections functions, including their motivation and construction, are presented in Ref. 36. Here only a brief summary is presented).…”

Section: B Extension Of Fr Approach To Trianglesmentioning

confidence: 99%

“…Note that for a single scheme, multiple vector correction functions are possible. 36 For the p = 2 case, only one of these possible functions is shown. The properties of a scheme are determined by the form of the correction field (φ f,j ), and not by the particular form of the vector correction function (h f,j ).…”

Section: B Extension Of Fr Approach To Trianglesmentioning

confidence: 99%

“…VCJH schemes specify a stability-preserving form for the correction functions and fields. 35,36 VCJH schemes for quadrilaterals (via 1D tensor products) and triangles are presented in the following sections.…”

Section: Vincent-castonguay-jameson-huynh Schemesmentioning

confidence: 99%

“…36 However, unique values for the coefficients of the polynomials defining each correction field φ f,j can be obtained from the solution of a system of equations. 36 Before, the aforementioned system of equations is presented, some notation must be introduced. First, let the operator D (v,w) be defined as…”

Section: Specifying the Correction Field φ Fjmentioning

confidence: 99%

See 4 more Smart Citations

An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids

Williams

Castonguay

Vincent

et al. 2011

20th AIAA Computational Fluid Dynamics Conference

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This paper extends the high-order Flux Reconstruction (FR) approach to the treatment of non-linear diffusive fluxes on triangles. The FR approach for solving diffusion problems is reviewed on quadrilaterals and extended for triangles, allowing the treatment of mixed grids. In particular, this paper examines a subset of FR schemes, referred to as VincentCastonguay-Jameson-Huynh (VCJH) schemes, which are provably stable across all orders of accuracy for linear fluxes in first order systems. The correction fields of the VCJH schemes are shown to represent a family of lifting operators which are used to enforce inter-element continuity of the solution and the diffusive flux. For diffusion problems, the lifting operators of nodal DG schemes are shown to be a subset of this family. Finally, numerical results are used to show the effectiveness of VCJH schemes for a range of problems, including the model diffusion equation and the compressible Navier-Stokes equations. Optimal orders of accuracy are obtained on unstructured mixed meshes of triangular and quadrilateral elements.

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Section: B Extension Of Fr Approach To Trianglesmentioning

confidence: 99%

“…(Full details of the vector corrections functions, including their motivation and construction, are presented in Ref. 36. Here only a brief summary is presented).…”

Section: B Extension Of Fr Approach To Trianglesmentioning

confidence: 99%

Section: B Extension Of Fr Approach To Trianglesmentioning

confidence: 99%

Section: Vincent-castonguay-jameson-huynh Schemesmentioning

confidence: 99%

Section: Specifying the Correction Field φ Fjmentioning

confidence: 99%

See 3 more Smart Citations

An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids

Williams

Castonguay

Vincent

et al. 2011

20th AIAA Computational Fluid Dynamics Conference

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Spectral Element andhpMethods

Kirby

Karniadakis

2017

Encyclopedia of Computational Mechanics Second Edition

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Spectral/ hp element methods provide high‐order discretization, which is essential in the longtime integration of advection–diffusion systems and for capturing dynamic instabilities in solids. In this chapter, we review the main formulations for simulations of incompressible and compressible viscous flows as well as for solid mechanics and present several examples with some emphasis on fluid–structure interactions and interfaces. The first generation of (nodal) spectral elements was limited to relatively simple geometries and smooth solutions. However, the new generation of spectral/ hp elements, consisting of both nodal and modal forms, can handle very complex geometries using unstructured grids and can capture strong shocks by employing discontinuous Galerkin methods. New flexible formulations allow simulations of multiphysics problems including extremely complex geometries and multiphase flows. Several implementation strategies have also been developed on the basis of multilevel parallel algorithms that allow dynamic ‐refinement at constant wall clock time. After three decades of intense developments, spectral element and hp methods are mature and efficient to be used effectively in applications of industrial complexity. They provide the capabilities that standard finite element and finite volume methods do, but, in addition, they exhibit high‐order accuracy and error control.

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Discrete filter operators for large‐eddy simulation using high‐order spectral difference methods

Lodato¹,

Castonguay

Jameson

2012

Numerical Methods in Fluids

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SUMMARYThe combination of a high‐order unstructured spectral difference (SD) spatial discretization scheme with sub‐grid scale (SGS) modeling for large‐eddy simulation is investigated with particular focus on the consistent implementation of a structural mixed model based on the scale similarity hypothesis. The difficult task of deriving a consistent formulation for the discrete filter within the SD element of arbitrary order led to the development of a new class of three‐dimensional constrained discrete filters. The discrete filters satisfy a set of selected criteria and are completely local within the SD element. Their weights can be automatically computed at run time from the number of solution points within each element and the expected filter cutoff length scale. The novel discrete filters can be applied to any SGS model involving explicit filtering and to a broad class of high‐order discontinuous finite element numerical schemes. The code is applied to the computation of turbulent channel flows at three Reynolds numbers, namely Reτ = 180, 395, and 590 (based on the friction velocity uτ and channel half‐width δ). Results from computations with and without the SGS model are compared against results from direct numerical simulation. The numerical experiments suggest that the results are sensitive to the use of the SGS model, even when a high‐order numerical scheme is used, especially when the grid resolution is kept relatively low and mostly in terms of resolved Reynolds stresses. Results obtained using existing filters based on the projection of the solution over lower‐order polynomial bases are also shown and demonstrate that these filters are inadequate for SGS modeling purposes, mostly because of their inability to enforce the selected cutoff length scale with sufficient accuracy. The use of the similarity mixed formulation proved to be particularly accurate in reproducing SGS interactions, confirming that its well‐known potential can be realized in conjunction with state‐of‐the‐art high‐order numerical schemes.Copyright © 2012 John Wiley & Sons, Ltd.

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A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements

Cited by 141 publications

References 24 publications

An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids

An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids

Spectral Element andhpMethods

Discrete filter operators for large‐eddy simulation using high‐order spectral difference methods

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