Upwind Schemes for the Navier-Stokes Equations

Koren, Barry

doi:10.1007/978-3-322-87869-4_31

Cited by 9 publications

(6 citation statements)

References 11 publications

(12 reference statements)

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“…gi+l/z,j = 9 (~~1 / 2 , j > u&l/2,j) hi,j+1/2 = h(u<j+l/2, ulj+i/z) and MUSCL extrapolation formulas In addition, Koren, 1988 has constructed the limiter…”

Section: Utilizing Monotone Lipschitz Continuous Numerical Flux Functmentioning

confidence: 99%

Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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“…gi+l/z,j = 9 (~~1 / 2 , j > u&l/2,j) hi,j+1/2 = h(u<j+l/2, ulj+i/z) and MUSCL extrapolation formulas In addition, Koren, 1988 has constructed the limiter…”

Section: Utilizing Monotone Lipschitz Continuous Numerical Flux Functmentioning

confidence: 99%

Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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“…(ii) Flux-vector splitting is less accurate than flux-difference splitting because it ignores contact discontinuities and shear waves. This inaccuracy particularly shows up as excessive numerical diffusion across a boundary layer [9,10]. Direct comparisons between Roe's and Osher's splittings are not known, but the methods should be very close in accuracy.…”

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confidence: 99%

Choice of implicit and explicit operators for the upwind differencing method

Liou

Leer

1988

26th Aerospace Sciences Meeting

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SUMMARYThe flux-vector and flux-difference splittings of Steger-Warming, Van Leer and Roe are tested in all possible combinations in the implicit and explicit operators that can be distinguished in implicit relaxation methods for the steady Euler and Navier-Stokes equations. The tests include one-dimensional inviscid nozzle flow, and two-dimensional inviscid and viscous shock reflection. Roe's splitting, as anticipated, is found to uniformly yield the most accurate results. On the other hand, an approximate Roe splitting of the implicit operator (the complete Roe splitting is too complicated for practical use) proves to be the least robust with regard to convergence to the steady state. In this respect, the Steger-Warming splitting is the most robust: it leads to convergence when combined with any of the splittings in the explicit operator, although not necessarily in the most efficient way.

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“…We use the limiter function of Koren [15,16] with the modification proposed by Venkatakrishnan [27], which can be written for the ξ direction as…”

Section: Governing Equations and Numerical Methodsmentioning

confidence: 99%

“…Fluxdifference splitting [22] has become very popular, often in combination with a third-order upwind-biased operator and a flux limiter [15,16]. Allmaras [1] used grid convergence studies to demonstrate the improvement in accuracy associated with the use of a fluxdifference-split upwind scheme over the scalar artificial dissipation scheme of Jameson et al [12] in the computation of laminar boundary layers.…”

Section: Introductionmentioning

confidence: 99%

Comparison of several spatial discretizations for the Navier-Stokes equations

Zingg¹,

Rango²,

Nemec³

et al. 1999

14th Computational Fluid Dynamics Conference

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Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier-Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourthorder differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.

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Upwind Schemes for the Navier-Stokes Equations

Cited by 9 publications

References 11 publications

Finite Volume Methods: Foundation and Analysis

Finite Volume Methods: Foundation and Analysis

Choice of implicit and explicit operators for the upwind differencing method

Comparison of several spatial discretizations for the Navier-Stokes equations

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