An Improved Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

Borges, Rafael J.; Costa, Bruno R. da; Don, Wai Sun

doi:10.21236/ada458949

Cited by 28 publications

(96 citation statements)

References 6 publications

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“…In [2,3], the improved performance of the WENO-Z5 scheme over the WENO-JS5 scheme has been demonstrated conclusively in computing a higher resolution solution when solving hyperbolic conservation laws for both smooth and discontinuous solutions. In [4], the WENO-Z5 scheme has been shown, in comparison to the WENO-JS5 scheme, to allow a more flexible choice on the as a function of grid spacing x that will guarantee the formal order of convergence of the nonlinear scheme regardless of the critical points while maintaining the essentially non-oscillatory nature of the shock-capturing scheme.…”

Section: Remarkmentioning

confidence: 96%

“…The smoothness of a given substencil is measured by the smoothness indicators {β k } r −1 k=0 that measure the normalized Sobolev norm of all the derivatives of each local polynomial. The nonlinear weights ω k , such as the classical weights given in the classical WENO-JS scheme [8] and the optimal weights given in the improved WENO-Z scheme [2,3], used in the formation of the convex combination of the local polynomials at the cell interface, have been used extensively. It has been numerically demonstrated that the WENO finite difference scheme based on the improved WENO-Z scheme is less dissipative and has a higher resolution power than the classical WENO-JS scheme for a larger class of problems.…”

Section: Introductionmentioning

confidence: 99%

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A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

2014

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The dissipation and dispersion (spectral) properties of the nonlinear fifth order classical weighted essentially non-oscillatory finite difference scheme (WENO-JS5) and its improved version (WENO-Z5) using the approximate dispersion relation (ADR) (Pirozzoli in J Comput Phys 219: [489][490][491][492][493][494][495][496][497] 2006) and the nonlinear spectral analysis (NSA) (Fauconnier et al. in J Comput Phys 228(6):1830-1861, 2009) are studied. Unlike the previous studies, the influences of the sensitivity parameter in the definition of the WENO nonlinear weights are also included for completeness. The fifth order upwinded central linear scheme (UW5) serves as the reference and benchmark for the purpose of comparison. The spectral properties of the WENO differentiation operator is well predicted theoretically by the ADR and validated numerically by the simulations of the WENO schemes in solving the scalar linear advection equation. In a long time simulation with an initial broadband wave, the WENO schemes generate spurious high modes with amplitude and spread of wavenumbers depend on the value of the sensitivity parameter. The NSA is applied to investigate the statistical nonlinear behavior, due to the nonlinear stencils adaptation of the WENO schemes, with a large set of initial conditions consisting of synthetic scalar fields with a prescribed energy spectrum and random phases. The statistics indicate that there is a small probability of an existence of a mild anti-dissipation in the low wavenumber range regardless of the size of the sensitivity parameter. Numerical examples demonstrate that the WENO-Z5 scheme is not only less dissipative and dispersive but also less sensitive to random phases than the WENO-JS5 scheme. Furthermore, a sensitivity parameter adaptive technique, in which its value depends 123 J Sci Comput on the local smoothness of the solution at a given spatial location and time, is introduced for solving a linear advection problem with a discontinuous initial condition. The preliminary result shows that the solution computed by the sensitivity parameter adaptive WENO-Z5 scheme agrees well with those computed by the WENO-Z5 scheme and the UW5 scheme in regions containing discontinuities and smooth solutions, respectively.

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Section: Remarkmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

2014

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“…For a better accuracy, an improved fifth-order WENO method is used for reconstructing the left and right state values [11] . In the method, a high-order spatial accuracy is achieved by means of a reconstruction procedure, which avoids generating spurious oscillations near the discontinuities and at the same time maintains the uniformly high-order accuracy in the regions where the solution is smooth.…”

Section: High-order Spacial Discretizationmentioning

confidence: 99%

“…Therefore, this paper focuses on the use of an improved fifth-order spatially accurate weighted ENO (WENO) scheme [11] for capturing the vortices while minimizing the numerical dissipation. The WENO scheme proposed by Liu et al [12] and extended by Jiang and Shu (WENO-JS) [13] can obtain good convergence properties while keeping the robustness and uniformly high-order accuracy of the ENO schemes.…”

Section: Introductionmentioning

confidence: 99%

“…The WENO scheme proposed by Liu et al [12] and extended by Jiang and Shu (WENO-JS) [13] can obtain good convergence properties while keeping the robustness and uniformly high-order accuracy of the ENO schemes. Borges et al [11] developed the WENO-JS scheme by the novel use of a linear combination of the low-order smoothness indicators already presented in the framework of the WENO-JS scheme, and provided a new WENO (WENO-Z) scheme with less dissipation and higher resolution than the classical WENO scheme. In an attempt to better capture the tip vortex and its trajectory, in the present work, the fifth-order WENO-Z scheme is reconstructed to interpolate the higher-order left and right states across a cell interface.…”

Section: Introductionmentioning

confidence: 99%

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Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

Weng

2011

Appl. Math. Mech.-Engl. Ed.

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A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws (MUSCL). For profitably capturing the wake and enforcing the period boundary condition, the computation regions of flows are discretized by using the structured chimera grids composed of a fine rotor grid and a cylindrical background grid. In the background grid, the mesh cells located in the wake regions are refined after the solution reaches the approximate convergence. Considering the interpolation characteristic of the WENO scheme, three layers of the hole boundary and the interpolation boundary are searched. The performance of the schemes is investigated in a transonic flow and a subsonic flow around the hovering rotor. The results reveal that the present approach has great capabilities in capturing the vortex wake with high resolution, and the WENO scheme has much lower numerical dissipation in comparison with the MUSCL scheme.

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Constrained moving least‐squares immersed boundary method for fluid‐structure interaction analysis

Batra

2017

Numerical Methods in Fluids

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Summary A numerical method is presented for the analysis of interactions of inviscid and compressible flows with arbitrarily shaped stationary or moving rigid solids. The fluid equations are solved on a fixed rectangular Cartesian grid by using a higher‐order finite difference method based on the fifth‐order WENO scheme. A constrained moving least‐squares sharp interface method is proposed to enforce the Neumann‐type boundary conditions on the fluid‐solid interface by using a penalty term, while the Dirichlet boundary conditions are directly enforced. The solution of the fluid flow and the solid motion equations is advanced in time by staggerly using, respectively, the third‐order Runge‐Kutta and the implicit Newmark integration schemes. The stability and the robustness of the proposed method have been demonstrated by analyzing 5 challenging problems. For these problems, the numerical results have been found to agree well with their analytical and numerical solutions available in the literature. Effects of the support domain size and values assigned to the penalty parameter on the stability and the accuracy of the present method are also discussed.

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An Improved Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

Cited by 28 publications

References 6 publications

A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

Constrained moving least‐squares immersed boundary method for fluid‐structure interaction analysis

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