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A Sixth-Order Weighted Essentially Non-oscillatory Schemes Based on Exponential Polynomials for Hamilton–Jacobi Equations

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Abstract

In this study, we present a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving Hamilton–Jacobi equations. The proposed scheme recovers the maximal approximation order in smooth regions without loss of accuracy at critical points. We incorporate exponential polynomials into the scheme to obtain better approximation near steep gradients without spurious oscillations. In order to design nonlinear weights based on exponential polynomials, we suggest an alternative approach to construct Lagrange-type exponential functions reproducing the cell-average values of exponential basis functions. Using the Lagrange-type exponential functions, we provide a detailed analysis of the approximation order of the proposed WENO scheme. Compared to other WENO schemes, the proposed scheme is simpler to implement, yielding better approximations with lower computational costs. A number of numerical experiments are presented to demonstrate the performance of the proposed scheme.

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Acknowledgements

Youngsoo Ha was supported by the grant NRF-2017R1D1A1B03034912 through the National Research Foundation of Korea and Chang Ho Kim was supported by Konkuk University. Jungho Yoon was supported by the grant NRF-2015R1A5A1009350 through the National Research Foundation of Korea and the MOTIE 10048720 through the Ministry of Trade, Industry and Energy of Korea.

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Authors and Affiliations

  1. Department of Mathematics, Seoul National University, Seoul, South Korea

    Youngsoo Ha

  2. Department of Computer Engineering, Glocal Campus, Konkuk University, Chungju, 380-701, South Korea

    Chang Ho Kim

  3. Department of Mathematics, Ewha W. University, Seoul, 120-750, South Korea

    Hyoseon Yang & Jungho Yoon

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  1. Youngsoo Ha
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  2. Chang Ho Kim
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Correspondence to Hyoseon Yang.

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Ha, Y., Kim, C.H., Yang, H. et al. A Sixth-Order Weighted Essentially Non-oscillatory Schemes Based on Exponential Polynomials for Hamilton–Jacobi Equations. J Sci Comput 75, 1675–1700 (2018). https://doi.org/10.1007/s10915-017-0603-8

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