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

Zhizhuang Zhang, Xiangyu Zhou, Gang Li (), Shouguo Qian and Qiang Niu
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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

Mathematics, 2023, vol. 11, issue 12, 1-18

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.

Keywords: hyperbolic conservation laws; finite difference scheme; entropy stable scheme; high-order accuracy (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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