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Discontinuity-Detecting Method for a Four-Point Stencil and Its Application to Develop a Third-Order Hybrid-WENO Scheme

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Abstract

First this paper analyzes the reason for the accuracy losing of the third-order weighted essentially non-oscillatory (WENO) scheme. It is shown that one reason is that the local smoothness indicators of the third-order WENO scheme cannot correctly treat the smooth three-point stencil containing a non-nodal critical point, here, ‘non-nodal’ means the critical point is not a grid point. And then a discontinuity-detecting method for a four-point stencil is proposed and applied to develop the high order accurate hybrid-WENO scheme by combining the third-order WENO scheme and a third-order upstream scheme. This four-point stencil is actually the stencil used for constructing the third-order WENO scheme (positive and negative numerical fluxes), hence the resulting hybrid-WENO scheme proposed by this paper does not introduce new grid point. Numerical examples show that the detecting method and the hybrid scheme are robust for problems with shocks, and the hybrid scheme obtains real third-order convergence rate for smooth solutions containing critical points.

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Acknowledgements

This research work was supported by the NKRDPC 2016YFA0401200, SCP No. TZ2016002, NSAF U1530145, NSFC Nos. 11872067 and 91852203.

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Authors and Affiliations

  1. State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Shengping Liu & Yiqing Shen

  2. School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China

    Shengping Liu

  3. School of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yiqing Shen

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  1. Shengping Liu
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  2. Yiqing Shen
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This research work was supported by the NKRDPC 2016YFA0401200, SCP No. TZ2016002, NSAF U1530145, NSFC Nos. 11872067 and 91852203.

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Liu, S., Shen, Y. Discontinuity-Detecting Method for a Four-Point Stencil and Its Application to Develop a Third-Order Hybrid-WENO Scheme. J Sci Comput 81, 1732–1766 (2019). https://doi.org/10.1007/s10915-019-01060-8

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  • DOI: https://doi.org/10.1007/s10915-019-01060-8

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