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A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws

Author

Listed:
  • Zhizhuang Zhang

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Xiangyu Zhou

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Gang Li

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Shouguo Qian

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Qiang Niu

    (Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China)

Abstract

The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical schemes cannot directly maintain discrete entropy inequalities. To address this, we here construct an ES finite difference scheme for the nonlinear hyperbolic systems of conservation laws. The proposed scheme can not only maintain the discrete entropy inequality, but also enjoy high-order accuracy. Firstly, we construct the second-order accurate semi-discrete entropy conservative (EC) schemes and ensure that the schemes meet the entropy identity when an entropy pair is given. Then, the second-order EC schemes are used as a building block to achieve the high-order accurate semi-discrete EC schemes. Thirdly, we add a dissipation term to the above schemes to obtain the high-order ES schemes. The term is based on the Weighted Essentially Non-Oscillatory (WENO) reconstruction. Finally, we integrate the scheme using the third-order Runge–Kutta (RK) approach in time. In the end, plentiful one- and two-dimensional examples are implemented to validate the capability of the scheme. In summary, the current scheme has sharp discontinuity transitions and keeps the genuine high-order accuracy for smooth solutions. Compared to the standard WENO schemes, the current scheme can achieve higher resolution.

Suggested Citation

  • Zhizhuang Zhang & Xiangyu Zhou & Gang Li & Shouguo Qian & Qiang Niu, 2023. "A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws," Mathematics, MDPI, vol. 11(12), pages 1-18, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2604-:d:1165824
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    References listed on IDEAS

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    1. Gassner, Gregor J. & Winters, Andrew R. & Kopriva, David A., 2016. "A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 291-308.
    2. Li, Gang & Li, Jiaojiao & Qian, Shouguo & Gao, Jinmei, 2021. "A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    3. Yuan, Xu-hua, 2018. "A well-balanced element-free Galerkin method for the nonlinear shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 46-53.
    4. Li, Gang & Caleffi, Valerio & Qi, Zhengkun, 2015. "A well-balanced finite difference WENO scheme for shallow water flow model," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1-16.
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