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A technique of treating negative weights in WENO schemes
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This paper presents a simple and effective technique for handling negative linear weights without a need to get rid of them and test cases are shown to illustrate the stability and accuracy of this approach.Abstract:
High-order accurate weighted essentially nonoscillatory (WENO) schemes have recently been developed for finite difference and finite volume methods both in structured and in unstructured meshes. A key idea in WENO scheme is a linear combination of lower order fluxes or reconstructions to obtain a higher order approximation. The combination coefficients, also called linear weights, are determined by local geometry of the mesh and order of accuracy and may become negative, such as in the central WENO schemes using staggered meshes, high-order finite volume WENO schemes in two space dimensions, and finite difference WENO approximations for second derivatives. WENO procedures cannot be applied directly to obtain a stable scheme if negative linear weights are present. The previous strategy for handling this difficulty is either by regrouping of stencils or by reducing the order of accuracy to get rid of the negative linear weights. In this paper we present a simple and effective technique for handling negative linear weights without a need to get rid of them. Test cases are shown to illustrate the stability and accuracy of this approach.read more
Citations
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Book ChapterDOI
Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws
TL;DR: In this paper, the authors describe the construction, analysis, and application of ENO and WENO schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations, where the key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible.
Journal ArticleDOI
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems
TL;DR: The history and basic formulation of WENO schemes are reviewed, the main ideas in using WenO schemes to solve various hyperbolic PDEs and other convection dominated problems are outlined, and a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications are presented.
Journal ArticleDOI
On maximum-principle-satisfying high order schemes for scalar conservation laws
Xiangxiong Zhang,Chi-Wang Shu +1 more
TL;DR: It is shown that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressibles velocity field.
Journal ArticleDOI
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems
Michael Dumbser,Martin Käser +1 more
TL;DR: A new WENO reconstruction technique is proposed that does not reconstruct point-values but entire polynomials which can easily be evaluated and differentiated at any point and thus can be implemented very efficiently even for unstructured grids in three space dimensions.
Journal ArticleDOI
High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD
TL;DR: Three types of high order methods being used in CFD are reviewed, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods.
References
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Journal ArticleDOI
Efficient Implementation of Weighted ENO Schemes
Guang-Shan Jiang,Chi-Wang Shu +1 more
TL;DR: A new way of measuring the smoothness of a numerical solution is proposed, emulating the idea of minimizing the total variation of the approximation, which results in a fifth-order WENO scheme for the caser= 3, instead of the fourth-order with the original smoothness measurement by Liuet al.
Journal ArticleDOI
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Chi-Wang Shu,Stanley Osher +1 more
TL;DR: Two methods of sharpening contact discontinuities-the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors originally used in the cell average framework-are applied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations.
Journal ArticleDOI
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
ShuChi-Wang,OsherStanley +1 more
TL;DR: This work extends earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock-capturing schemes by providing a new simplified expression for the ENO constructio...
Journal ArticleDOI
Weighted essentially non-oscillatory schemes
TL;DR: A new version of ENO (essentially non-oscillatory) shock-capturing schemes which is called weighted ENO, where, instead of choosing the "smoothest" stencil to pick one interpolating polynomial for the ENO reconstruction, a convex combination of all candidates is used.
Journal ArticleDOI
Uniformly high order accurate essentially non-oscillatory schemes, 111
TL;DR: An hierarchy of uniformly high-order accurate schemes is presented which generalizes Godunov's scheme and its second- order accurate MUSCL extension to an arbitrary order of accuracy.
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