Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points

Henrick, Andrew; Aslam, Tariq D.; Powers, Joseph M.

doi:10.1016/j.jcp.2005.01.023

Cited by 788 publications

(821 citation statements)

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“…For more details about the scheme and solver, see Toro (1999). It is worth mentioning that for simulations of compressible flow with discontinuities using shock-capturing numerical schemes, convergence rates are often found to be at most first order, even if nominally high-order methods are used in smooth regions of the flow (Carpenter & Casper 1992;Yamaleev & Carpenter 1995;Henrick, Aslam & Powers 2005). To achieve global high accuracy (or convergence rate) of numerical methods, higher-order schemes combined with shock-fitting technique are being developed so as to avoid the corrupting influences of numerical viscosity introduced in common shock-capturing methods (e.g.…”

Section: Methodsmentioning

confidence: 99%

Numerical study on unstable surfaces of oblique detonations

2014

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In this study, the onset of cellular structure on oblique detonation surfaces is investigated numerically using a one-step irreversible Arrhenius reaction kinetic model. Two types of oblique detonations are observed from the simulations. One is weakly unstable characterized by the existence of a planar surface, and the other is strongly unstable characterized by the immediate formation of the cellular structure. It is found that a high degree of overdrive suppresses the formation of cellular structures as confirmed by the results of many previous studies. However, the present investigation demonstrates that cellular structures also appear with degree of overdrive of 2.06 and 2.37, values much higher than ∼1.8 suggested previously in the literature for the critical value defining the instability boundary of oblique detonations. This contradiction could be explained by the use of differently shaped walls, a straight wall used in this study and a custom-designed curved wedge system so as to induce straight oblique detonations in previous studies. Another possible reason could be due to the low and possibly insufficient resolution used in previously published studies. Hence, simulations with different grid sizes are also performed to examine the effect of resolution on the numerical solutions. Using the present results, analysis also shows that although the characteristic lengths of unstable surfaces are different when the incident Mach number changes, these length scales are proportional to tangential velocities. Hence, the interior time determined by the overdrive degree is identified, and its limitation as the instability parameter is discussed.

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Section: Methodsmentioning

confidence: 99%

Numerical study on unstable surfaces of oblique detonations

2014

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“…This dissipation can be reduced by hybridizing the WENO method with a high-order scheme, and the resolving power can be improved by optimizing the stencil with a compact centraldifference scheme in smooth flow regions [22]. Mapped WENO schemes were developed to improve the accuracy at critical points where derivatives vanish [23]. A hybrid fifth-order compact upwind-WENO scheme was developed for shock-turbulence interaction [24].…”

Section: Benefits Of the Weno Methods For Simulating Complex Shocked Fmentioning

confidence: 99%

Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer–Meshkov instability

Latini

Schilling

Don

2007

Journal of Computational Physics

119

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Weighted essentially non-oscillatory (WENO) simulations of the reshocked twodimensional single-mode Richtmyer-Meshkov instability using third-, fifth-and ninthorder spatial flux reconstruction and uniform grid resolutions corresponding to 128, 256 and 512 points per initial perturbation wavelength are presented. The dependence of the density, vorticity, simulated density Schlieren and baroclinic production fields, mixing layer width, circulation deposition, mixing profiles, production and mixing fractions, energy spectra, statistics, probability distribution functions, numerical turbulent kinetic energy and enstrophy production/dissipation rates, numerical Reynolds numbers, and numerical viscosity on the order and resolution is investigated to long evolution times. The results are interpreted using the implicit numerical dissipation in the characteristic projection-based, finite-difference WENO method. It is shown that higher order higher resolution simulations have lower numerical dissipation. The sensitivity of the quantities considered to the order and resolution is further amplified following reshock, when the energy deposition by the second shock-interface interaction induces the formation of small-scale structures. Lower-order lower-resolution simulations preserve large-scale structures and flow symmetry to late times, while higher-order higher-resolution simulations exhibit fragmentation of the structures, symmetry breaking and increased mixing. Similar flow features are qualitatively and quantitatively captured by either approximately doubling the order or the resolution. Additionally, the computational scaling shows that increasing the order is more advantageous than increasing the resolution for the flow considered here. The present investigation suggests that the ninth-order WENO method is well-suited for the simulation and analysis of complex multi-scale flows and mixing generated by shock-induced hydrodynamic instabilities.

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“…Through the novel use of higher order information already present in the framework of the classical scheme, new smoothness indicators are devised and we obtain a new WENO scheme with less dissipation than the classical WENO of Jiang and Shu [2], with the same computational cost, and a slightly better performance than the improved mapped version of Henrick et al [3]. We show that the enhancements of the new scheme come from its ability to assign substantially larger weights to discontinuous stencils than the previous versions of WENO.…”

mentioning

confidence: 99%

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

Borges¹,

Costa²,

Don³

2006

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We develop in this article an improved version of the fifth-order weighted essentially non-oscillatory (WENO) scheme. Through the novel use of higher order information already present in the framework of the classical scheme, new smoothness indicators are devised and we obtain a new WENO scheme with less dissipation than the classical WENO of Jiang and Shu [2], with the same computational cost, and a slightly better performance than the improved mapped version of Henrick et al [3]. We show that the enhancements of the new scheme come from its ability to assign substantially larger weights to discontinuous stencils than the previous versions of WENO. Numerical experiments with the linear advection of discontinuous functions and the one dimensional Euler system of equations are conducted to demonstrate the benefit of using this improved version of the WENO scheme for hyperbolic conservation laws.

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Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points

Cited by 788 publications

References 20 publications

Numerical study on unstable surfaces of oblique detonations

Numerical study on unstable surfaces of oblique detonations

Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer–Meshkov instability

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

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