Comparison of two formulations for high-order accurate essentially nonoscillatory schemes

Casper, Jay; Shu, Chi‐Wang; Atkins, Harold L.

doi:10.2514/3.12240

Cited by 45 publications

(9 citation statements)

References 12 publications

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“…This discrepancy in cost is even bigger for three dimensions. We refer to [18] for a detailed comparison of multi-dimensional finite volume and finite difference schemes in the context of ENO reconstructions. Notice that this difference between finite volume and finite difference schemes is meaningful only for schemes of at least third order accuracy.…”

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confidence: 99%

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

Shu¹

2009

SIAM Rev.

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806

589

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High order accurate weighted essentially non-oscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, non-oscillatory and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the users to tune parameters, thus they are very convenient to use for practitioners. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic partial differential equations and other convection dominated problems, and present a collected sample of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, and computational biology. Finally, we mention a few topics currently being investigated about WENO schemes.

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confidence: 99%

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

Shu¹

2009

SIAM Rev.

Self Cite

806

589

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“…However, in multidimensions, finite volume schemes are much more costly than finite difference schemes, since a reconstruction from multi-dimensional cell averages to point values needs to be performed and numerical quadratures must be used for the integration to get numerical fluxes. We refer to [204,25,270] for the discussion of multi-dimensional finite volume schemes and their comparison with the finite difference counter-parts. However, finite volume schemes have a significant advantage over finite difference schemes, in that they can be designed on nonsmooth and unstructured meshes without losing their high order accuracy and conservation properties, see, for example, [1,72,103,66,283] for high order finite volume ENO and 6 WENO schemes on two-and three-dimensional unstructured meshes.…”

Section: Finite Difference and Finite Volume Weno Schemesmentioning

confidence: 99%

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Shu

2016

Journal of Computational Physics

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164

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For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax-Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.

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“…The two-dimensional channel flow is an isentropic flow governed by the Euler equations, which was first discussed by Casper et al [35]. In this paper, the C3 geometry is used, and the shape of the middle section is determined by…”

Section: Two-dimensional Channel Flowmentioning

confidence: 99%

Effect of Nonuniform Grids on High-Order Finite Difference Method

Deng

Chen

et al. 2017

Adv. Appl. Math. Mech.

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The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

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Comparison of two formulations for high-order accurate essentially nonoscillatory schemes

Cited by 45 publications

References 12 publications

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Effect of Nonuniform Grids on High-Order Finite Difference Method

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