A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws

Zhu, Jun; Qiu, Jianxian

doi:10.1016/j.jcp.2016.05.010

Cited by 233 publications

(146 citation statements)

References 22 publications

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“…We use the implicit large eddy simulation methodology [5,[17][18][19][20][21][22] where weighted non-oscillatory (WENO) upwinding schemes [23][24][25] are coupled with certain Riemann solvers in an implicit large eddy simulation (or ILES) framework. The term implicit refers to the addition of numerical dissipation by the upwinding scheme rather than any explicit dissipation term [26].…”

Section: Introductionmentioning

confidence: 99%

Resolution and Energy Dissipation Characteristics of Implicit LES and Explicit Filtering Models for Compressible Turbulence

Maulik

San

2017

Fluids

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Solving two-dimensional compressible turbulence problems up to a resolution of 16, 384 2 , this paper investigates the characteristics of two promising computational approaches: (i) an implicit or numerical large eddy simulation (ILES) framework using an upwind-biased fifth-order weighted essentially non-oscillatory (WENO) reconstruction algorithm equipped with several Riemann solvers, and (ii) a central sixth-order reconstruction framework combined with various linear and nonlinear explicit low-pass spatial filtering processes. Our primary aim is to quantify the dissipative behavior, resolution characteristics, shock capturing ability and computational expenditure for each approach utilizing a systematic analysis with respect to its modeling parameters or parameterizations. The relative advantages and disadvantages of both approaches are addressed for solving a stratified Kelvin-Helmholtz instability shear layer problem as well as a canonical Riemann problem with the interaction of four shocks. The comparisons are both qualitative and quantitative, using visualizations of the spatial structure of the flow and energy spectra, respectively. We observe that the central scheme, with relaxation filtering, offers a competitive approach to ILES and is much more computationally efficient than WENO-based schemes.

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Section: Introductionmentioning

confidence: 99%

Resolution and Energy Dissipation Characteristics of Implicit LES and Explicit Filtering Models for Compressible Turbulence

Maulik

San

2017

Fluids

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“…The fifth-order finite volume SWENO scheme will be used to numerically solve the spatial derivatives of shallow water equations [32,37]. In this section, the basic procedure of SWENO will be described as short overview for the following 1D scalar hyperbolic conservation law:…”

Section: The Fifth-ordermentioning

confidence: 99%

“…In 2016, a successful type of WENO [32] was proposed to approximate hyperbolic conservation laws. The new WENO reconstruction is a convex combination of two linear polynomials with a fourth-degree polynomial using the same five-point big stencil as in the classic WENO fashion; the linear weights are constants, which are positive, and their summation equals one so they are more simple and easily extended to multidimensions in engineering applications.…”

Section: Introductionmentioning

confidence: 99%

“…In the scheme we follow the ideas in [32,37] about simple finite volume WENO scheme and the algebraic prebalance method for the flux and the source terms [36,38] in shallow water equations. An outline of the paper is as follows: in Section 2 we describe details of the discretization with finite volume SWENO scheme and Lax-Wendroff-type time discretization for shallow water equations.…”

Section: Introductionmentioning

confidence: 99%

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The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

Xie

Yang

2018

Mathematical Problems in Engineering

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A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.

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“…In this paper, following the idea of the new type WENO schemes [34,35,9,36,37], hybrid WENO [22,14,4,5,20,38] and hybrid HWENO [39], we develop the new hybrid HWENO scheme in which we use a nonlinear convex combination of a high degree polynomial with several low degree polynomials and the linear weights can be any artificial positive numbers with the only constraint that their sum is one. The procedures of the new hybrid HWENO scheme are: firstly, we modify the first order moments using the new HWENO limiter methodology in the troubled-cells, which are identified by the KXRCF troubled-cell indicator [16].…”

Section: Introductionmentioning

confidence: 99%

A hybrid Hermite WENO scheme for hyperbolic conservation laws

Zhao

Chen

Qiu

2020

Journal of Computational Physics

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In this paper, a fifth-order Hermite weighted essentially non-oscillatory (HWENO) scheme with artificial linear weights is proposed for one and two dimensional hyperbolic conservation laws, where the zeroth-order and the first-order moments are used in the spatial reconstruction. We construct the HWENO methodology using a nonlinear convex combination of a high degree polynomial with several low degree polynomials, and the associated linear weights can be any artificial positive numbers with only requirement that their summation equals one.The one advantage of the HWENO scheme is its simplicity and easy extension to multidimension in engineering applications for we can use any artificial linear weights which are independent on geometry of mesh. The another advantage is its higher order numerical accuracy using less candidate stencils for two dimensional problems. In addition, the HWENO scheme still keeps the compactness as only immediate neighbor information is needed in the reconstruction and has high efficiency for directly using linear approximation in the smooth regions. In order to avoid nonphysical oscillations nearby strong shocks or contact discontinuities, we adopt the thought of limiter for discontinuous Galerkin method to control the spurious oscillations. Some benchmark numerical tests are performed to demonstrate the capability of the proposed scheme.

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A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws

Cited by 233 publications

References 22 publications

Resolution and Energy Dissipation Characteristics of Implicit LES and Explicit Filtering Models for Compressible Turbulence

Resolution and Energy Dissipation Characteristics of Implicit LES and Explicit Filtering Models for Compressible Turbulence

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

A hybrid Hermite WENO scheme for hyperbolic conservation laws

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