高阶保号熵稳定格式

数学物理学报 ›› 2021, Vol. 41 ›› Issue (5): 1296-1310.

高阶保号熵稳定格式

郑素佩(),徐霞*(),封建湖(),贾豆()

^{长安大学理学院西安 710064}

收稿日期:2020-05-27 出版日期:2021-10-26 发布日期:2021-10-08
通讯作者: 徐霞 E-mail:zsp2008@chd.edu.cn;xuxiachina@163.com;jhfeng@chd.edu.cn;1429594854@qq.com
作者简介:郑素佩, E-mail: zsp2008@chd.edu.cn|封建湖, E-mail: jhfeng@chd.edu.cn|贾豆, E-mail: 1429594854@qq.com
基金资助:
国家自然科学基金(11971075);陕西省自然科学基金(2020JQ-338);陕西省自然科学基金(2019JM-243)

High Order Sign Preserving Entropy Stable Schemes

Supei Zheng(),Xia Xu*(),Jianhu Feng(),Dou Jia()

^{College of Science, Chang'an University, Xi'an 710064}

Received:2020-05-27 Online:2021-10-26 Published:2021-10-08
Contact: Xia Xu E-mail:zsp2008@chd.edu.cn;xuxiachina@163.com;jhfeng@chd.edu.cn;1429594854@qq.com
Supported by:
the NSFC(11971075);the NSF of Shaanxi Province(2020JQ-338);the NSF of Shaanxi Province(2019JM-243)

摘要/Abstract

摘要：

高阶熵稳定格式构造的一个核心任务是如何保证熵变量在进行高阶重构前后的符号不变.该文构造了高阶保号熵稳定格式（熵守恒通量采用Fjordholm方案，耗散部分的熵变量采用三阶紧致CWENO重构），证明了基于该重构的熵变量在跳跃间断处满足保号性.数值结果表明，该格式达到三阶精度、分辨率高、鲁棒性强且无振荡产生.

关键词: 保号性, 紧致CWENO重构, 熵稳定格式

Abstract:

Ensuring the sign preserving on entropy variable after the high-order reconstruction is fundamental in constructing the high-order entropy stable schemes. In this paper, we construct the 3rd order compact CWENO-type entropy stable schemes (Fjordholm's the 3rd order entropy conservative schemes with sign preserving compact CWENO reconstruction for the entropy variable) and prove the sign preserving on entropy variable with the 3rd order compact CWENO reconstruction. Numerical results show that the schemes can achieve third-order accuracy, and have high resolution, robustness and non-oscillation.

Key words: Sign preserving, Compact CWENO reconstruction, Entropy stable schemes

中图分类号:

O354

郑素佩,徐霞,封建湖,贾豆. 高阶保号熵稳定格式[J]. 数学物理学报, 2021, 41(5): 1296-1310.

Supei Zheng,Xia Xu,Jianhu Feng,Dou Jia. High Order Sign Preserving Entropy Stable Schemes[J]. Acta mathematica scientia,Series A, 2021, 41(5): 1296-1310.

参考文献

1	Lax P D . Weak solutions of nonlinear hyperbolic equation and their numerical computation. Commun Pur Appl Math, 1954, 7 (1): 159- 193
2	Lax P D . Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Reg Conf Ser Appl Math, 1973, 11, 1- 48
3	Tadmor E . The numerical viscosity of entropy stable schemes for systems of conservation laws. I Math Comp, 1987, 49 (179): 91- 103
4	Lefloch P G , Mercier J M , Rohde C . Fully discrete, entropy conservative schemes of arbitrary order. SIAM J Numer Anal, 2002, 40 (5): 1968- 1992
5	Cheng X , Nie Y . A third-order entropy stable scheme for hyperbolic conservation laws. J Hyperbol Differ Eq, 2016, 13 (1): 129- 145
6	Fjordholm U S , Mishra S , Tadmor E . Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J Numer Anal, 2012, 50 (2): 544- 573
7	Fjordholm U S , Ray D . A sign preserving WENO reconstruction method. SIAM J Sci Comput, 2015, 68 (1): 42- 63
8	Biswas B , Dubey R K . Low dissipative entropy stable schemes using third order WENO and TVD reconstructions. Adv Comput Math, 2018, 44 (4): 1153- 1181
9	Ismail F , Roe P L . Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J Comput Phys, 2009, 228 (15): 5410- 5436
10	Levy D , Puppo G , Russo G . Compact central WENO schemes for multidimensional conservation laws. SIAM J Sci Comput, 2000, 22 (2): 656- 672
11	Jameson A . Analysis and design of numerical schemes for gas dynamics, 1:Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. Comput Fluids, 1995, 4 (3/4): 171- 218
12	Gottlieb S , Shu C W . Total variation diminishing Runge-Kutta schemes. Math Comput, 1998, 67 (221): 73- 85
13	Zakerzadeh H , Fjordholm U S . High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA J Numer Anal, 2016, 36 (2): 633- 654
14	Jameson A . The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. J Sci Comput, 2008, 34 (2): 152- 187
15	Dehghan M , Jazlanian R . On the total variation of a third-order semi-discrete central scheme for 1D conservation laws. J Vib Control, 2011, 17 (9): 1348- 1358
16	Puppo G A . Numerical entropy production for central schemes. SIAM J Sci Comput, 2003, 25 (4): 1382- 415
17	Tadmor E . Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer, 2003, 12, 451- 512
18	Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws//Quarteroni A. Advanced Numer Appr Nonli Hyper Equa. Berlin: Springer-Verlag, 2006: 325-432
19	Fjordholm U S , Mishra S , Tadmor E . Energy preserving and energy stable schemes for the shallow water equations. Found Comput Math, 2009, 363 (14): 93- 139
20	陈雨风, 陈停停, 王振. 非等熵Chaplygin气体测度值解存在性. 数学物理学报, 2020, 40A (4): 833- 841
20	Chen Y F , Chen T T , Wang Z . The existence of the measure solution for the non-isentropic chaplygin gas. Acta Math Sci, 2020, 40A (4): 833- 841
21	陈停停, 屈爱芳, 王振. 等熵Chaplygin气体的二维Riemann问题. 数学物理学报, 2017, 37A (6): 1053- 1061
21	Chen T T , Qu A F , Wang Z . The two-dimensional riemann problem for isentropic chaplygin gas. Acta Math Sci, 2017, 37A (6): 1053- 1061
22	吴宏伟. 可压缩磁流体动力方程解的正则性. 数学物理学报, 2010, 30A (3): 593- 602
22	Wu H W . Regularity criteria for the compressible magneto-hydrodynamic equations. Acta Math Sci, 2010, 30A (3): 593- 602
23	Tadmor E . Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discret Contin Dyn syst, 2016, 36 (8): 4579- 4598