A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws
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 num...
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/12/2604 |
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| author | Zhizhuang Zhang Xiangyu Zhou Gang Li Shouguo Qian Qiang Niu |
| author_facet | Zhizhuang Zhang Xiangyu Zhou Gang Li Shouguo Qian Qiang Niu |
| author_sort | Zhizhuang Zhang |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T02:12:09Z |
| format | Article |
| id | doaj.art-294a75f0a02a409b8d922162039fb442 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T02:12:09Z |
| publishDate | 2023-06-01 |
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| spelling | doaj.art-294a75f0a02a409b8d922162039fb4422023-11-18T11:27:14ZengMDPI AGMathematics2227-73902023-06-011112260410.3390/math11122604A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation LawsZhizhuang Zhang0Xiangyu Zhou1Gang Li2Shouguo Qian3Qiang Niu4School of Mathematics and Statistics, Qingdao University, Qingdao 266071, ChinaSchool of Mathematics and Statistics, Qingdao University, Qingdao 266071, ChinaSchool of Mathematics and Statistics, Qingdao University, Qingdao 266071, ChinaSchool of Mathematics and Statistics, Qingdao University, Qingdao 266071, ChinaDepartment of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou 215123, ChinaThe 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.https://www.mdpi.com/2227-7390/11/12/2604hyperbolic conservation lawsfinite difference schemeentropy stable schemehigh-order accuracy |
| spellingShingle | Zhizhuang Zhang Xiangyu Zhou Gang Li Shouguo Qian Qiang Niu A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws Mathematics hyperbolic conservation laws finite difference scheme entropy stable scheme high-order accuracy |
| title | A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws |
| title_full | A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws |
| title_fullStr | A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws |
| title_full_unstemmed | A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws |
| title_short | A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws |
| title_sort | new entropy stable finite difference scheme for hyperbolic systems of conservation laws |
| topic | hyperbolic conservation laws finite difference scheme entropy stable scheme high-order accuracy |
| url | https://www.mdpi.com/2227-7390/11/12/2604 |
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