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

Vincent, Peter; Castonguay, Patrice; Jameson, Antony

doi:10.1007/s10915-010-9420-z

Cited by 360 publications

(293 citation statements)

References 13 publications

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“…Setting c > 0 was shown to have a stabilizing effect compared to the baseline case of c = 0 by Vincent et al [14].…”

mentioning

confidence: 87%

“…The FR framework is easy to implement on unstructured meshes, allows for tuning of the numerical properties and is cheap to compute owing to the lack of integration procedures. A family of Energy-Stable Flux Reconstruction (ESFR) schemes have been developed by the Aerospace Computing Lab (ACL) at Stanford University [14]. The ESFR schemes were proven to be stable in an energy norm of Sobolev type for the 1D linear advection equation for all orders of accuracy on an arbitrary mesh [14].…”

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confidence: 99%

“…A family of Energy-Stable Flux Reconstruction (ESFR) schemes have been developed by the Aerospace Computing Lab (ACL) at Stanford University [14]. The ESFR schemes were proven to be stable in an energy norm of Sobolev type for the 1D linear advection equation for all orders of accuracy on an arbitrary mesh [14]. Hence the ESFR schemes can be formulated on quadrilateral and hexahedral elements by taking tensor products of 1D operators.…”

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confidence: 99%

“…The energy norm contains a non-negative coefficient c which allows for the recovery of particular collocation-based nDG (with c = 0) and SD (c = c SD ) schemes -henceforth denoted as FR-nDG and FR-SD resp. -as well as the G 2 scheme (c = c G2 ) of Huynh and an infinite variety of new stable schemes [14].…”

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confidence: 99%

“…Upwinding with a Roe flux was shown to improve the stability of the ESFR schemes (compared to central flux) in 1D by Vincent et al [14]. It was shown by Jameson and Lodato [31] that, for the ESFR schemes, the amount of dissipation added by the interface flux is proportional to a high-order derivative of the solution, thus providing an efficient damping of the energy in the high-order modes.…”

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confidence: 99%

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Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

Bull

Jameson

2015

AIAA Journal

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“…Setting c > 0 was shown to have a stabilizing effect compared to the baseline case of c = 0 by Vincent et al [14].…”

mentioning

confidence: 87%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

Bull

Jameson

2015

AIAA Journal

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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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Shock capturing with the high‐order flux reconstruction method on adaptive meshes based on p4est

Xia

2022

Engineering Reports

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High order schemes have been investigated for quite a long time, and the flux reconstruction (FR) scheme proposed by Huynh recently attracts the attention of researchers due to its simplicity and efficiency. Building the framework that bridges discontinuous Galerkin (DG) and spectral difference (SD) schemes, FR recovers DG and SD conveniently with a careful selection of parameters. In this article, FR scheme is realized based on the framework of p4est, an open source adaptive mesh refinement library. The shock capturing ability of localized Laplacian artificial viscosity and in‐cell piecewise integrated solution methods are compared. Curved boundary treatment for high order schemes is adopted. The performance of developed code is estimated in both one and two dimensions including curved boundary and shock cases, and some attractive results are obtained.

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

Cited by 360 publications

References 13 publications

Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

Spectral Element andhpMethods

Shock capturing with the high‐order flux reconstruction method on adaptive meshes based on p4est

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