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An oscillation-free Hermite WENO scheme for hyperbolic conservation laws

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Abstract

In this paper, the sixth-order oscillation-free Hermite weighted essentially non-oscillatory (OF-HWENO) scheme is proposed for hyperbolic conservation laws on structured meshes, where the zeroth- and first-order moments are the variables for the governing equations. The main difference from other HWENO schemes existed in the literature is that we add high-order numerical damping terms in the first-order moment equations to control spurious oscillations for the OF-HWENO scheme. The OF-HWENO scheme not only can achieve the designed optimal numerical order, but also can be easily implemented as we use only one set of stencil in the reconstruction procedure and the same reconstructed polynomials are applied for the zeroth- and first-order moments equations. In order to obtain the adaptive order resolution when facing the discontinuities, a transition polynomial is added in the reconstruction, where the associated linear weights can also be any positive numbers as long as their summation equals one. In addition, the OF-HWENO scheme still keeps the compactness as only immediate neighbor values are needed in the space discretization. Some benchmark numerical tests are performed to illustrate the high-order accuracy, high resolution and robustness of the proposed scheme.

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Acknowledgements

Jianxian Qiu was supported by National Key R&D Program of China (Grant No. 2022YFA1004501). Zhuang Zhao was supported by the Postdoctoral Science Foundation of China (Grant No. 2021M702145). The authors thank Chair Professor Shi Jin at Shanghai Jiao Tong University for his helpful comments on the paper, and the first author Zhuang Zhao also thanks PhD Candidates Jiayin Li and Chuan Fan at Xiamen University for helpful discussions in the numerical implementation.

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  1. School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Zhuang Zhao

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Jianxian Qiu

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  1. Zhuang Zhao
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Correspondence to Jianxian Qiu.

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Zhao, Z., Qiu, J. An oscillation-free Hermite WENO scheme for hyperbolic conservation laws. Sci. China Math. 67, 431–454 (2024). https://doi.org/10.1007/s11425-022-2064-1

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  • DOI: https://doi.org/10.1007/s11425-022-2064-1

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