A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements

Balan, Aravind; May, Georg; Schöberl, Joachim

doi:10.1016/j.jcp.2011.11.041

Cited by 40 publications

(37 citation statements)

References 21 publications

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“…Here d is the dimension of the problem, u( x, t) is the conserved variable, f (u( x, t)) is the vector of the flux function and x is the coordinate in the physical domain with components x i , i = 1, ..., d. Note we reserve vector notation for the flux function like [36,37],…”

Section: Brief Review Of the Sd Methodsmentioning

confidence: 99%

A new spectral difference method using hierarchical polynomial bases for hyperbolic conservation laws

Xie

Zhang

et al. 2015

Journal of Computational Physics

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Section: Brief Review Of the Sd Methodsmentioning

confidence: 99%

A new spectral difference method using hierarchical polynomial bases for hyperbolic conservation laws

Xie

Zhang

et al. 2015

Journal of Computational Physics

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“…Later, the introduction of a new basis function for the flux appears to have fixed the problem [50].…”

Section: (B) Related Formulationsmentioning

confidence: 99%

High-order computational fluid dynamics tools for aircraft design

Wang

2014

Phil. Trans. R. Soc. A.

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Most forecasts predict an annual airline traffic growth rate between 4.5 and 5% in the foreseeable future. To sustain that growth, the environmental impact of aircraft cannot be ignored. Future aircraft must have much better fuel economy, dramatically less greenhouse gas emissions and noise, in addition to better performance. Many technical breakthroughs must take place to achieve the aggressive environmental goals set up by governments in North America and Europe. One of these breakthroughs will be physics-based, highly accurate and efficient computational fluid dynamics and aeroacoustics tools capable of predicting complex flows over the entire flight envelope and through an aircraft engine, and computing aircraft noise. Some of these flows are dominated by unsteady vortices of disparate scales, often highly turbulent, and they call for higher-order methods. As these tools will be integral components of a multi-disciplinary optimization environment, they must be efficient to impact design. Ultimately, the accuracy, efficiency, robustness, scalability and geometric flexibility will determine which methods will be adopted in the design process. This article explores these aspects and identifies pacing items.

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“…Note that placements of these points are staggered to enhance stability of the approximation to hyperbolic conservation laws. Stability of SD methods is investigated in [1,10].…”

Section: Introductionmentioning

confidence: 99%

“…The hybrid difference is composed of two kinds of finite difference approximations as follows. The 5-point stencil FD approximation of the Poisson equation, for example, at the cell centered point η is the cell finite difference: 1) and the FD approximation of the flux continuity at η is the interface finite difference:…”

Section: Introductionmentioning

confidence: 99%

Hybrid Spectral Difference Methods for an Elliptic Equation

Jeon

Park

Shin

2017

Computational Methods in Applied Mathematics

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A locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, resulting in the system of equations in the cell interface unknowns only. A complete ellipticity analysis is provided. The optimal order of convergence in the semi-discrete energy norms is proved. Several numerical results are given to show the performance of the method, which confirm our theoretical findings.

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A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements

Cited by 40 publications

References 21 publications

A new spectral difference method using hierarchical polynomial bases for hyperbolic conservation laws

A new spectral difference method using hierarchical polynomial bases for hyperbolic conservation laws

High-order computational fluid dynamics tools for aircraft design

Hybrid Spectral Difference Methods for an Elliptic Equation

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