A novel and robust scale-invariant WENO scheme for hyperbolic conservation laws

Published on Jan 1, 2022in Journal of Computational Physics3.553
· DOI :10.1016/J.JCP.2021.110724
Wai Sun Don24
Estimated H-index: 24
(Ocean University of China),
Run Li1
Estimated H-index: 1
(Ocean University of China)
+ 1 AuthorsYinghua Wang2
Estimated H-index: 2
(Nanjing Tech University)
Source
Abstract
Abstract null null A novel, simple, robust, and effective modification in the nonlinear weights of the scale-invariant WENO operator is proposed that achieves an optimal order of accuracy with smooth function regardless of the critical point (Cp-property), a scale-invariant with an arbitrary scaling of a function (Si-property), an essentially non-oscillatory approximation of a discontinuous function (ENO-property), and, in some cases, a well-balanced WENO finite difference/volume scheme (WB-property) (up to machine rounding error numerically). The classical WENO-JS/Z/D operators do not satisfy the Si-property intrinsically due to a loss of sub-stencils' adaptivity in the WENO reconstruction of a discontinuous function when scaled by a small scaling factor. By introducing the descaling function, an average of the function values in the weights to build the scale-invariant WENO-JSm/Zm/Dm operators, the operators are independent of both the scaling factor and sensitivity parameter. The Si-property and Cp-property of the WENO operators are validated theoretically and numerically in quadruple-precision with small and large scaling factors and sensitivity parameters. The results show that the WENO-JSm/Zm/Dm operators satisfy the Si-property and the WENO-D/Dm operators satisfy the Cp-property. Furthermore, the ENO-property of the WENO-Zm/Dm schemes is illustrated via several one- and two-dimensional shock-tube problems. In solving the Euler equations under gravitational fields, the well-balanced scale-invariant WENO schemes achieve the WB-property intrinsically without imposing the stringent homogenization condition.
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References30
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#1Peng LiH-Index: 5
#2Bao-Shan Wang (Ocean University of China)H-Index: 4
Last. Wai Sun Don (Ocean University of China)H-Index: 24
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Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. Li and Xing (J Comput Phys 316:145–163, 2016) proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium....
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#1Peng LiH-Index: 5
#2Zhen Gao (Ocean University of China)H-Index: 11
Abstract The compressible Euler equations coupled with the gravitational source terms admit a hydrostatic equilibrium state where the gradients of the flux terms can be exactly balanced by those in the source terms. This property of exact preservation of the equilibrium is highly desirable at the discrete level when they are numerically solved. In this study, we design the simple high order well-balanced finite difference weighted essentially non-oscillatory (WENO-R/I) schemes, which base on the...
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#1Peng LiH-Index: 5
#2Wai Sun Don (Ocean University of China)H-Index: 24
Last. Zhen Gao (Ocean University of China)H-Index: 11
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Abstract A numerical framework of the generalized form of high order well-balanced finite difference weighted essentially non-oscillatory (WENO) interpolation-based schemes is proposed for the shallow water equations. It demonstrates more flexible construction process than the classical WENO reconstruction-based schemes. The weighted compact nonlinear schemes and finite difference alternative WENO schemes are two specific cases. To maintain the exact C-property, the splitting technique for the s...
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#1Yiqing ShenH-Index: 1
#2Ke ZhangH-Index: 1
Last. Jun PengH-Index: 1
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A novel method for constructing robust and high-order accurate weighted essentially non-oscillatory (WENO) scheme is proposed in this paper. The method is mainly based on the WENO-Z type scheme, in which, an eighth-order global smoothness indicator (the square of the approximation of the fourth-order derivative on the five-point stencil used by the fifth-order WENO scheme) is used, and in order to keep the ENO property and robustness, the constant 1 used to calculate the un-normalized weights is...
#1Manuel J. Castro (UMA: University of Málaga)H-Index: 28
#2Carlos Parés (UMA: University of Málaga)H-Index: 24
In some previous works, the authors have introduced a strategy to develop well-balanced high-order numerical methods for nonconservative hyperbolic systems in the framework of path-conservative numerical methods. The key ingredient of these methods is a well-balanced reconstruction operator, i.e. an operator that preserves the stationary solutions in some sense. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-bal...
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#1Yinghua Wang (Ocean University of China)H-Index: 2
#2Bao-Shan Wang (Ocean University of China)H-Index: 4
Last. Wai Sun Don (Ocean University of China)H-Index: 24
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A modified fifth order Z-type (nonlinear) weights, which consist of a linear term and a nonlinear term, in the weighted essentially non-oscillatory (WENO) polynomial reconstruction procedure for the WENO-Z finite difference scheme in solving hyperbolic conservation laws is proposed. The nonlinear term is modified by a modifier function that is based on the linear combination of the local smoothness indicators. The WENO scheme with the modified Z-type weights (WENO-D) scheme and its improved vers...
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#1Antonio BaezaH-Index: 24
#2Raimund BürgerH-Index: 35
Last. David ZoríoH-Index: 7
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A modified weighted essentially nonoscillatory (WENO) reconstruction technique preventing accuracy loss near critical points (regardless of their order) of the underlying data is presented. This ap...
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This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is kno...
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Abstract A high-order well-balanced scheme for the Euler equations with gravitation is presented. The scheme is able to preserve a spatially high-order accurate discrete representation of isentropic hydrostatic equilibria. It is based on a novel local hydrostatic reconstruction, which, in combination with any standard high-order accurate reconstruction procedure, achieves genuine high-order accuracy for smooth solutions close or away from equilibrium. The resulting scheme is very simple and can ...
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#1Jun Zhu (NUAA: Nanjing University of Aeronautics and Astronautics)H-Index: 16
#2Chi-Wang Shu (Brown University)H-Index: 105
Abstract In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25] , [45] , could obtain the optimal order of accu...
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