Towards optimal high-order compact schemes for simulating compressible flows

Zhang, Huaibao; Zhang, Fan; Xu, Chunguang

doi:10.1016/j.amc.2019.03.001

Cited by 19 publications

(5 citation statements)

References 45 publications

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“…As mentioned above, in order to calculate u L∕R,i± 1 2 , we proposed an improved fifth-order compact nonlinear scheme 25 using a stencil-selection procedure 23 in the nonlinear interpolation. This is carried out by first introducing a smoothness measurement technique followed by a cut-off procedure.…”

Section: Node-to-midpoint Interpolationmentioning

confidence: 99%

“…In the proposed work, we aim to develop a high-resolution third-order compact nonlinear scheme, since it is simple and efficient in practical applications. Recently, we incorporated the stencil-selection procedure 23,24 to improve WCNS, resulting fifth-order and sixth-order low-dissipation compact nonlinear scheme 25,26 within the framework of WCNS, exhibiting superior spectral properties and shock-capturing capabilities. The same concept of stencil-selection is applied in this work, yet a more efficient nonlinear function, which is built in the form of the JS weight, 19 is adopted for the present scheme to measure the smoothness of candidate stencils.…”

Section: Introductionmentioning

confidence: 99%

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A third‐order compact nonlinear scheme for compressible flow simulations

Tian

Wang

Zhang

et al. 2020

Numerical Methods in Fluids

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Summary Reynolds‐averaged Navier‐Stokes simulations based on second‐order numerical methods are widely used by commercial codes and work as dominating tools for most industrial applications. They, however, suffer from limitations in accurate and reliable predictions of skin‐friction drag and aerodynamic heating, as well as in simulations of complex flows such as large‐scale separation and transition. A remedy for this is the development of high‐order schemes, by which numerically induced dissipation and dispersion errors of low‐order schemes can be effectively reduced. Weighted compact nonlinear schemes (WCNSs) are a family of high‐resolution nonlinear shock‐capturing methods. A stencil‐selection procedure is introduced in the proposed work with an aim to improve the nonlinear weight of the third‐order WCNS. By using the approximate dispersion relation analysis, it is demonstrated that the new scheme has reduced dissipation and dispersion errors, compared with WCNSs using two typical nonlinear weights. Improvements are also achieved by the new scheme in numerical tests such as the double Mach reflection problem and the Rayleigh‐Taylor instability simulation, which are characterized by strong shock discontinuities and rich small scales, respectively. The new scheme is therefore highly favored in the simulation of flow problems involving strong discontinuities and multiscales phenomena.

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Section: Node-to-midpoint Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A third‐order compact nonlinear scheme for compressible flow simulations

Tian

Wang

Zhang

et al. 2020

Numerical Methods in Fluids

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“…However, the relatively large dispersion and dissipation errors of low-order schemes will smear out a lot of flow details in smooth regions [2], which has promoted the development of high-order nonlinear schemes. Representative achievements of high-order nonlinear schemes include ENO (essentially non-oscillatory) [3], WENO (weighted ENO) [4], TENO (targeted ENO) [5], CNS (compact nonlinear scheme) [6], WCNS (weighted CNS) [7], TCNS (targeted CNS) [8], etc. Nonlinear weighting has played an important role in developing some of these schemes, which provide largely improved scheme accuracy and computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

Improvement of Z-Weighted Function Based on Fifth-Order Nonlinear Multi-Order Weighted Method for Shock Capturing of Hyperbolic Conservation Laws

Bai,

Yan,

Mao

et al. 2024

Entropy

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Based on a 5-point stencil and three 3-point stencils, a nonlinear multi-order weighted method adaptive to 5-3-3-3 stencils for shock capturing is presented in this paper. The form of the weighting function is the same as JS (Jiang–Shu) weighting; however, the smoothness indicator of the 5-point stencil adopts a special design with a higher-order leading term similar to the τ in Z weighting. The design maintains that the nonlinear weights satisfy sufficient conditions for the scheme to avoid degradation even near extreme points. By adjusting the linear weights to a specific value and using the τ in Z weighting, the method can be degraded to Z weighting. Analysis of linear weights shows that they do not affect the accuracy in the smooth region, and they can also adjust the resolution and discontinuity-capturing capability. Numerical tests of different hyperbolic conservation laws are conducted to test the performance of the newly designed nonlinear weights based on the weighted compact nonlinear scheme. The numerical results show that there are no obvious oscillations near the discontinuity, and the resolution of both the discontinuity and smooth regions is better than that of Z weights.

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“…14 Despite the success of these high-order schemes, in order to increase the order of accuracy of a specific numerical scheme, usually, more degree-of-freedoms (DOFs) are necessary for constructing higher-order polynomials, and thus more computational effort is inevitable. For finite difference schemes, including WENO schemes 7,15,16 and WCNSs, [17][18][19] DOFs are increased by extending the candidate stencil(s). Balsara and Shu proposed a class of numerical schemes 15 that are higher-order extensions of the WENO-JS scheme.…”

Section: Introductionmentioning

confidence: 99%

Arbitrary high‐order extended essentially non‐oscillatory schemes for hyperbolic conservation laws

Zhang

Dong

et al. 2021

Numerical Methods in Fluids

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Achieving high numerical resolution in smooth regions and robustness near discontinuities within a unified framework is the major concern while developing numerical schemes solving hyperbolic conservation laws, for which the essentially non-oscillatory (ENO) type scheme is a favorable solution. Therefore, an arbitrary-high-order ENO-type framework is designed in this article. With using a typical five-point smoothness measurement as the shock-detector, the present schemes are able to detect discontinuities before spatial reconstructions, and thus more spatial information can be exploited to construct incremental-width stencils without crossing discontinuities, ensuring ENO property and high-order accuracy at the same time. The present shock-detection procedure is specifically examined for justifying its performance of resolving high-frequency waves, and a standard metric for discontinuous solutions is also applied for measuring the shock-capturing error of the present schemes, especially regarding the amplitude error in post-shock regions. In general, the present schemes provide high-resolution, and more importantly, the schemes are more efficient compared with the typical WENO schemes since only a five-point smoothness measurement is applied for arbitrary-high-order schemes. Numerical results of canonical test cases also provide evidences of the overall performance of the present schemes.

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Towards optimal high-order compact schemes for simulating compressible flows

Cited by 19 publications

References 45 publications

A third‐order compact nonlinear scheme for compressible flow simulations

A third‐order compact nonlinear scheme for compressible flow simulations

Improvement of Z-Weighted Function Based on Fifth-Order Nonlinear Multi-Order Weighted Method for Shock Capturing of Hyperbolic Conservation Laws

Arbitrary high‐order extended essentially non‐oscillatory schemes for hyperbolic conservation laws

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