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New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

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Abstract

In this paper, a new type of finite difference mapped weighted essentially non-oscillatory (MWENO) schemes with unequal-sized stencils, such as the seventh-order and ninth-order versions, is constructed for solving hyperbolic conservation laws. For the purpose of designing increasingly high-order finite difference WENO schemes, the equal-sized stencils are becoming more and more wider. The more we use wider candidate stencils, the bigger the probability of discontinuities lies in all stencils. Therefore, one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some four-point or five-point stencils into several smaller three-point stencils. By the usage of this new methodology in high-order spatial reconstruction procedure, we get different degree polynomials defined on these unequal-sized stencils, and calculate the linear weights, smoothness indicators, and nonlinear weights as specified in Jiang and Shu (J. Comput. Phys. 126: 202228, 1996). Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions, another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights, so as to keep the optimal order of accuracy in smooth regions. These new MWENO schemes can also be applied to compute some extreme examples, such as the double rarefaction wave problem, the Sedov blast wave problem, and the Leblanc problem with a normal CFL number. Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

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Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China

    Jun Zhu

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

    Jianxian Qiu

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  1. Jun Zhu
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Correspondence to Jianxian Qiu.

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Additional information

Jun Zhu: Research was supported by the NSFC grant 11872210 and the Science Challenge Project, No. TZ2016002. The author was also partly supported by the NSFC Grant 11926103 when he visited Tianyuan Mathematical Center in Southeast China, Xiamen 361005, Fujian, China.

Jianxian Qiu: Research was supported by the NSFC Grant 12071392 and the Science Challenge Project, No. TZ2016002.

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Zhu, J., Qiu, J. New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy. Commun. Appl. Math. Comput. 5, 64–96 (2023). https://doi.org/10.1007/s42967-021-00122-9

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