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152
Efficient Implementation of Weighted ENO Schemes
, 1995
"... In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially nonoscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order accur ..."
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Cited by 412 (38 self)
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In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially nonoscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order
HighOrder ENO Schemes for Unstructured Meshes Based on LeastSquares Reconstruction
 AIAA Paper 970540
, 1997
"... This article is part of an ongoing effort to develop highorder schemes for unstructured meshes to the point where meaningful information can be obtained about the tradeoffs involved in using spatial discretizations of higher than secondorder accuracy on unstructured meshes. This article describes ..."
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Cited by 6 (2 self)
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describes a highorder accurate ENO reconstruction scheme, called DDL 2 ENO, for use with vertexcentered upwind flow solution algorithms on unstructured meshes. The solution of conservation equations in this context can be broken naturally into three phases: 1. Solution reconstruction, in which a
Data Bounded Polynomials and Preserving Positivity in High Order ENO and WENO Methods
, 2009
"... The positivity and accuracy properties of the widely used ENO and WENO methods are considered by undertaking an analysis based upon databounded polynomial methods. The positivity preserving approach of Berzins based upon databounded polynomial interpolants is generalized to arbitrary meshes. This ..."
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Cited by 2 (2 self)
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. This makes it possible to prove positivity conditions for ENO Methods by using a derivation based on such bounded polynomial approximations. Numerical examples are used to show that although high order methods may be used in a way that preserves positivity, care must be taken in terms of resolving shock
A high order ENO conservative Lagrangian scheme for the compressible Euler equations
"... and ChiWang Shu2 We develop a class of Lagrangian schemes for solving the Euler equations of compressible gas dynamics both in the Cartesian and in the cylindrical coordinates. The schemes are based on high order essentially nonoscillatory (ENO) reconstruction. They are conservative for the densit ..."
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and ChiWang Shu2 We develop a class of Lagrangian schemes for solving the Euler equations of compressible gas dynamics both in the Cartesian and in the cylindrical coordinates. The schemes are based on high order essentially nonoscillatory (ENO) reconstruction. They are conservative
Weighted ENO Schemes for HamiltonJacobi Equations
 SIAM J. Sci. Comput
, 1997
"... In this paper, we present a weighted ENO (essentially nonoscillatory) scheme to approximate the viscosity solution of the HamiltonJacobi equation: OE t +H(x 1 ; \Delta \Delta \Delta ; x d ; t; OE; OE x1 ; \Delta \Delta \Delta ; OE xd ) = 0: This weighted ENO scheme is constructed upon and has the ..."
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Cited by 229 (0 self)
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the same stencil nodes as the 3 rd order ENO scheme but can be as high as 5 th order accurate in the smooth part of the solution. In addition to the accuracy improvement, numerical comparisons between the two schemes also demonstrate that, the weighted ENO scheme is more robust than the ENO scheme. Key
Weighted Essentially NonOscillatory Schemes
, 1994
"... In this paper we introduce a new version of ENO (Essentially NonOscillatory) shockcapturing schemes which we call Weighted ENO. The main new idea is that, instead of choosing the "smoothest" stencil to pick one interpolating polynomial for the ENO reconstruction, we use a convex combinati ..."
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Cited by 326 (8 self)
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combination of all candidates to achieve the essentially nonoscillatory property, while additionally obtaining one order of improvement in accuracy. The resulting Weighted ENO schemes are based on cellaverages and a TVD RungeKutta time discretization. Preliminary encouraging numerical experiments are given.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 270 (26 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di
ENO reconstruction and ENO interpolation are stable
 FOUND COMPUT MATH
, 2012
"... ... are stable in the sense that the jump of the reconstructed ENO point values at each cell interface has the same sign as the jump of the underlying cell averages across that interface. Moreover, we prove that the size of these jumps after reconstruction relative to the jump of the underlying cell ..."
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Cited by 4 (3 self)
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cell averages is bounded. Similar sign properties and the boundedness of the jumps hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on nonuniform meshes, indicate a remarkable rigidity
Entropy stable ENO scheme
, 2010
"... We prove the first stability estimates for the ENO reconstruction procedure. They take the form of a sign property: we show that the jump in the reconstructed pointvalues at each cell interface has the same sign as the jump in underlying cell averages (cell centered values) across the interface. M ..."
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. Moreover their ratio is upper bounded. These estimates hold for arbitrary orders of accuracy of the reconstruction as well as for nonuniform meshes. We then combine the ENO reconstruction together with entropy conservative fluxes to construct new entropy stable ENO schemes of arbitrary order.
Results 1  10
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152