A Flux Reconstruction Approach to High-Order Schemes  Including Discontinuous Galerkin Methods

Huynh, H. T.

doi:10.2514/6.2007-4079

Cited by 702 publications

(753 citation statements)

References 15 publications

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“…which are the left and right Radau polynomials respectively, hence c = 0 recovers a particular FRnDG scheme as shown by Huynh [13]. The recovered scheme uses a collocation projection of the flux onto a polynomial space of degree p, which has significant implications for the nonlinear stability.…”

Section: B Energy-stable Flux Reconstruction Schemesmentioning

confidence: 99%

“…Dissipation is reduced at higher p, which translates as better resolving efficiency. Figure 3 plots the real and imaginary components of the physical mode for FR-nDG, FR-SD and Huynh's G 2 scheme [13] at order p 3 . The effect of changing c from c = c nDG = 0 to c = c SD is that the numerical wavespeed remains closer to the exact wavespeed for longer, but the dissipation starts increasing (i.e.…”

Section: Spectral Propertiesmentioning

confidence: 99%

“…A third scheme, identified by Huynh as being particularly stable, is referred to as the G 2 scheme [13] and is recovered by choosing c = c G2 given by…”

Section: B Energy-stable Flux Reconstruction Schemesmentioning

confidence: 99%

“…F o r P e e r R e v i e w general high-order framework which, in the case of linear fluxes, recovers particular collocation-based nDG and SD methods as well as allowing for the definition of new schemes [13]. 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.…”

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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Section: B Energy-stable Flux Reconstruction Schemesmentioning

confidence: 99%

Section: Spectral Propertiesmentioning

confidence: 99%

“…A third scheme, identified by Huynh as being particularly stable, is referred to as the G 2 scheme [13] and is recovered by choosing c = c G2 given by…”

Section: B Energy-stable Flux Reconstruction Schemesmentioning

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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“…It has been shown previously that a weak imposition of well-posed boundary conditions for finite difference [7,8], finite volume [9,10], spectral element [11,12], discontinuous Galerkin [13,14] and flux reconstruction schemes [15,16] on summation-by-parts (SBP) form can lead to energy stability. We will show that the continuous analysis of well posed boundary conditions implemented with weak boundary procedures together with schemes on Summation-by-parts (SBP) form automatically leads to stability.…”

Section: Introductionmentioning

confidence: 99%

A Roadmap to Well Posed and Stable Problems in Computational Physics

Nordström

2016

J Sci Comput

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All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier-Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

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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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A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods

Cited by 702 publications

References 15 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

A Roadmap to Well Posed and Stable Problems in Computational Physics

Spectral Element andhpMethods

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