Improvement of third-order WENO-Z scheme at critical points

Xu, Wei; Wu, Weiguo

doi:10.1016/j.camwa.2018.02.009

Cited by 22 publications

(11 citation statements)

References 22 publications

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“…In addition, L ∞ numerical order of WENO3‐L3 also converges to third‐order, while that of WENO3‐L4 only converge to 2.5. Similar performance on L ∞ numerical order reduction were also shown in References 15,20.Example linear problem with a discontinuity.…”

Section: Numerical Testssupporting

confidence: 73%

“…One is to set the parameter ϵ in Equation (12) greater than or equal to the square of the mesh size △ x 2 ( M1‐type ), as in References 13, 17. The other is to take a power of τ or IS k ( M2‐type ), as in References 14–17, 20. However, both of them would cause some problems owing to the choice of reference values in the nondimensionalizing process.…”

Section: Numerical Testsmentioning

confidence: 99%

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Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

Guo

Sun

et al. 2020

Numerical Methods in Fluids

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In this article, we present two improved third-order weighted essentially nonoscillatory (WENO) schemes for recovering their design-order near first-order critical points. The schemes are constructed in the framework of third-order WENO-Z scheme. Two new global smoothness indicators, τ L3 and τ L4 , are devised by a nonlinear combination of local smoothness indicators (IS k) and reference values (IS G) based on Lagrangian interpolation polynomial. The performances of the proposed schemes are evaluated on several numerical tests governed by one-dimensional linear advection equation or oneand two-dimensional Euler equations. Numerical results indicate that the presented schemes provide less dissipation and higher resolution than the original WENO3-JS and subsequent WENO3-N scheme.

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Section: Numerical Testssupporting

confidence: 73%

Section: Numerical Testsmentioning

confidence: 99%

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

Guo

Sun

et al. 2020

Numerical Methods in Fluids

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“…The function α k of WENO‐Z3 defined as

\begin{align} α_{k} & = d_{k} ((), 1 + \frac{τ_{z}}{β_{k} + ε}), k = 0, 1, \end{align}

\begin{align} τ_{z} & = false| β_{0} prefix- β_{1} false|, ε = 1 0^{prefix- 40}, \end{align}

where τ z is the global smoothness indicator. Now, we review the accuracy of the WENO‐Z3 scheme with the help of Taylor expansion 16,17 . In order to prove the convergence order of the proposed scheme (WENO‐L3) at critical points, the Taylor expansion of the smoothness indicators is extended to O ( h 8 ) here.…”

Section: Methodsmentioning

confidence: 99%

“…Xu and Wu 16 added a new term to the weights of the WENO‐N3 scheme to further slightly increase the weight of less‐smooth stencil and obtained a new scheme (WENO‐P3). Xu et al 17 presented an improved third‐order WENO scheme (WENO‐PZ3) by slightly modifying the local smoothness indicators of the conventional WENO‐Z scheme. At the same time, Xu et al 18 proposed another improved WENO scheme (WENO‐NN3) to recover the convergence order at critical points and the global smoothness indicator is cited from Reference 13.…”

Section: Introductionmentioning

confidence: 99%

An improved third‐order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws

et al. 2020

Numerical Methods in Fluids

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Summary In this article, we present an improved third‐order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO‐ZQ scheme. However, the global smoothness indicator has a little different from WENO‐ZQ scheme. In this follow‐up article, a convex combination of a second‐degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three‐point information is adopted by the improved third‐order WENO scheme, the truncation errors are smaller than some other third‐order WENO schemes in L∞ and L2 norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one‐ and two‐dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third‐order WENO schemes.

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“…Therefore, and inspiring by [24,25], we propose new and simple smoothness indicator denoted asβ h (22), which can be used to replace β h in (20) andβ h in (21). The new smoothness indicator is constructed based on an idea of linearly combining the existing smoothness indicators of third-order linear reconstructions; this will lead to reduce the complexity of that of WENO-AO and WENO-AON schemes.…”

Section: A New and Simple Smoothness Indicatormentioning

confidence: 99%

A New Smoothness Indicator of Adaptive Order Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

2020

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Adaptive order weighted essentially non-oscillatory scheme (WENO-AO(5,3)) has increased the computational cost and complexity of the classic fifth-order WENO scheme by introducing a complicated smoothness indicator for fifth-order linear reconstruction. This smoothness indicator is based on convex combination of three third-order linear reconstructions and fifth-order linear reconstruction. Therefore, this paper proposes a new simple smoothness indicator for fifth-order linear reconstruction. The devised smoothness indicator linearly combines the existing smoothness indicators of third-order linear reconstructions, which reduces the complexity of that of WENO-AO(5,3) scheme. Then WENO-AO(5,3) scheme is modified to WENO-O scheme with new and simple formulation. Numerical experiments in 1-D and 2-D were run to demonstrate the accuracy and efficacy of the proposed scheme in which WENO-O scheme was compared with original WENO-AO(5,3) scheme along with WENO-AO-N, WENO-Z, and WENO-JS schemes. The results reveal that the proposed WENO-O scheme is not only comparable to the original scheme in terms of accuracy and efficacy but also decreases its computational cost and complexity.

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Improvement of third-order WENO-Z scheme at critical points

Cited by 22 publications

References 22 publications

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

An improved third‐order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws

A New Smoothness Indicator of Adaptive Order Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

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