A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions

Zhang, Shuhai; Shu, Chi‐Wang

doi:10.1007/s10915-006-9111-y

Cited by 101 publications

(89 citation statements)

References 21 publications

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“…The main difficulties lie on the non-differentiability of the numerical scheme leading to the creation of slight oscillations nearby shock discontinuities. More adapted WENO schemes then have been developed [52,23,21] to overcome the problem of convergence toward the steady-state solution. We also mention some interesting developments for the steady-state case based on the Discontinuous Galerkin method [6,7,38], the Spectral Volume technique [10,18], and the Residual Distribution schemes [12,2,22].…”

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confidence: 99%

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

2017

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To cite this version:Stéphane Clain, Raphaël Loubère, Gaspar Machado. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. 2016. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equationsWe propose a new family of finite volume high-accurate numerical schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers', and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

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confidence: 99%

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

2017

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“…In recent years, we have been studying the mechanism of sound generation in the interaction between a shock wave and a single planar vortex [14], a pair of planar vortices [15,16] or a longitudinal vortex [17] through direct numerical simulation (DNS) for the two and three dimensional unsteady compressible Navier-Stokes equations using the fifth order weighted essentially nonoscillatory (WENO) finite difference scheme developed by Jiang and Shu [18] and improved by Zhang and Shu [19,20] for the aeroacoustic computation. We have obtained many interesting findings, including the multi-stage interaction between a shock and a strong vortex [14] and limit cycles in the interaction between a normal shock wave and a longitudinal vortex [17].…”

Section: Security Classification Of: 17 Limitation Of Abstractmentioning

confidence: 99%

Sound Generation in the Interaction of Two Isentropic Vortices

Zhang¹,

Zhang²,

Shu³

2012

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Summary: Through direct numerical simulation (DNS) for the sound generated by the interaction between two isentropic vortices, it is found that the interaction between two vortices with a large difference in their strengths or scales can generate continuous strong noise. The interaction between two isentropic vortices results in the formation of two vortex dipoles, with each vortex dipole containing two vortex cores. If there is a large difference in their initial strengths, there is a large difference in the strengths of the resulted vortex cores. As a result, the weaker vortex rotates around the stronger one continuously, which is the key source of the aerodynamic sound. This DNS result provides a new concept: the interactions among turbulent structures with large differences in their scales play a key role in the generation of turbulent noise.Key Words: WENO scheme, direct numerical simulation, sound wave, isentropic vortex.

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“…In recent years, several variants and improvements of the original WENO schemes [14,15] have been proposed [20][21][22][23][24]. By introducing an anti-diffusive flux, Despres and Lagoutiere [20] proposed a first-order approach called limited downwind scheme to prevent the smearing of contact discontinuities while keeping nonlinear stability.…”

Section: Introductionmentioning

confidence: 99%

“…Also, a mapped WENO scheme has been devised by Henrick et al [23] that improves over the original WENO scheme at critical points. Zhang and Shu [24] suggested a modified smoothness indicator for the fifth-order WENO scheme near a steady shock region and they showed that the residue for the WENO scheme with the new smoothness indicator can converge to machine zero for one-and two-dimensional steady problems with strong shock wave when there is no influence from the boundary conditions. Recently, a numerical comparison of several variant WENO schemes for the compressible Euler equations has been given [25], which includes a lower-upper Gauss-Seidel implicit WENO scheme with anti-diffusive flux.…”

Section: Introductionmentioning

confidence: 99%

Simulation of multiple shock–shock interference using implicit anti‐diffusive WENO schemes

Hsieh

Wang

Yang

2009

Numerical Methods in Fluids

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SUMMARYAccurate computations of two-dimensional turbulent hypersonic shock-shock interactions that arise when single and dual shocks impinge on the bow shock in front of a cylinder are presented. The simulation methods used are a class of lower-upper symmetric-Gauss-Seidel implicit anti-diffusive weighted essentially non-oscillatory (WENO) schemes for solving the compressible Navier-Stokes equations with Spalart-Allmaras one-equation turbulence model. A numerical flux of WENO scheme with anti-diffusive flux correction is adopted, which consists of first-order and high-order fluxes and allows for a more flexible choice of first-order dissipative methods. Experimental flow fields of type IV shock-shock interactions with single and dual incident shocks by Wieting are computed. By using the WENO scheme with anti-diffusive flux corrections, the present solution indicates that good accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Computed surface pressure distribution and heat transfer rate are also compared with experimental data and other computational results and good agreement is found.

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A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions

Cited by 101 publications

References 21 publications

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

Sound Generation in the Interaction of Two Isentropic Vortices

Simulation of multiple shock–shock interference using implicit anti‐diffusive WENO schemes

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