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Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

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Abstract

High-order accurate weighted essentially non-oscillatory (WENO) schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations (PDEs). Due to highly nonlinear property of the WENO algorithm, large amount of computational costs are required for solving multidimensional problems. In our previous work (Lu et al. in Pure Appl Math Q 14: 57–86, 2018; Zhu and Zhang in J Sci Comput 87: 44, 2021), sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations, and it was shown that significant CPU times were saved, while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids. In this technical note, we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme, which has very interesting properties such as its simplicity in linear weights’ construction over a classical WENO scheme. Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times, and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

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Authors and Affiliations

  1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Ernie Tsybulnik, Xiaozhi Zhu & Yong-Tao Zhang

Authors
  1. Ernie Tsybulnik
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  2. Xiaozhi Zhu
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  3. Yong-Tao Zhang
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Correspondence to Yong-Tao Zhang.

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The authors declare that there is no conflict of interest.

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Research was partially supported by NSF Grant DMS-1620108.

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Tsybulnik, E., Zhu, X. & Zhang, YT. Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations. Commun. Appl. Math. Comput. 5, 1339–1364 (2023). https://doi.org/10.1007/s42967-022-00202-4

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  • DOI: https://doi.org/10.1007/s42967-022-00202-4

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