加热下分数阶广义二阶流体的Stokes第一问题的高阶数值方法

叶超; 骆先南; 文立平

doi:10.3879/j.issn.1000-0887.2012.01.006

加热下分数阶广义二阶流体的Stokes第一问题的高阶数值方法

doi: 10.3879/j.issn.1000-0887.2012.01.006

湘潭大学数学与计算科学学院, 湖南湘潭 411105

基金项目: 国家自然科学基金资助项目(10971175);湖南省教育厅重点科研资助项目(09A093)

详细信息

通讯作者:
叶超(1985—),男,湖北人,硕士(联系人.E-mail:yechaofan@gmail.com).

中图分类号: O241.82
计量
- 文章访问数: 1782
- HTML全文浏览量: 153
- PDF下载量: 851
- 被引次数: 0
出版历程
- 收稿日期: 2011-08-03
- 修回日期: 2011-11-07
- 刊出日期: 2012-01-15

High-Order Numerical Methods of the Fractional Order Stokes’ First Problem for a Heated Generalized Second Grade Fluid

School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P.R.China

摘要

摘要: 针对一类带Dirichlet边值条件和初值条件的加热下分数阶广义二阶流体的Stokes第一问题，提出了一种新的高阶隐式数值格式．应用Fourier分析方法和矩阵方法研究了该格式的稳定性、可解性及收敛性．也进一步给出一个时间误差阶更高的改进的隐式格式．最后通过两个数值算例验证了格式的理论分析是有效可靠的．
- 分数阶Stokes问题 /
- 隐式差分格式 /
- 可解性 /
- 稳定性 /
- 收敛性
Abstract: High-order implicit finite difference methods for solving the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative were studied. The stability, solvability and convergence of the numerical scheme were discussed via fourier analysis and matrix analysis method. An improved implicit scheme was also obtained. Finally, two numerical examples were presented to demonstrate the effectiveness of the mentioned schemes.
- fractional order Stokes’ first problem /
- implicit difference scheme /
- solvability stability /
- convergence /

HTML全文

参考文献(22)

[1]	Podlubny I. Fractional Differential Equations[M]. San Diego: Academic Press, 1999.
[2]	Shriram Srinivasana, Rajagopal K R. Study of a variant of Stokes’first and second problems for fluids with pressure dependent viscosities[J]. International Journal of Engineering Science, 2009, 47(11/12): 1357-1366.
[3]	Beiro da Veigaa L, Gyrya V, Lipnikov K, Manzini G. Mimetic finite difference method for the Stokes problem on polygonal meshes[J]. Journal of Computational Physics, 2009, 228(19): 7215-7232.
[4]	Ivan C Christov. Stokes’first problem for some non-Newtonian fluids: Results and mistakes[J]. Mechanics Research Communications, 2010, 37(8): 717-723.
[5]	庄平辉，刘青霞. 加热下分数阶广义二阶流体的Rayleigh-Stokes问题的一种有效数值方法[J]. 应用数学和力学， 2009, 30(12): 1440-1452.（ZHUANG Ping-hui, LIU Qing-xia. Numerical method of Rayleigh-Stokes problem for heated generalizedsecond grade fluid with fractional derivative[J]. Applied Mathematics and Mechanics(English Edition), 2009, 30(12): 1533-1546.
[6]	TAN Wen-chang, Masuoka Takashi. Stokes’first problem for a second grade fluid in a porous half-space with heated boundary[J]. International Journal of Non-Linear Mechanics, 2005, 40(4): 515-522.
[7]	Hayat T, Shahazad F, Ayub M. Stokes’first problem for the fourth order fluid in a porous half space[J]. Acta Mechanica Sinica, 2007, 23(1): 17-21.
[8]	Devakara M, Iyengar T K V. Stokes’ first problem for a micropolar fluid through state-space approach[J]. Applied Mathematical Modelling, 2009, 33(2): 924-936.
[9]	Salah Faisal, Zainal Abdul Aziz, Dennis Ling Chuan Ching. New exact solution for Rayleigh-Stokes problem of Maxwell fluid in a porous medium and rotating frame[J]. Results in Physics, 2011, 1(1): 9-12.
[10]	Magdy A. Ezzat, Hamdy M. Youssef. Stokes’first problem for an electro-conducting micropolar fluid with thermoelectric properties[J]. Canadian Journal of Physics, 2010, 88(1): 35-48.
[11]	SHEN Fang, TAN Wen-chang, ZHAO Yao-hua, Takashi Masuoka. The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model[J]. Nonlinear Analysis: Real World Applications, 2006, 7(5): 1072-1080.
[12]	CHEN Chang-ming, LIU F, Anh V. A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative[J]. J Comput Appl Math, 2009, 223(2): 777-789.
[13]	Liu F, Yang Q, Turner I. Stability and convergence of two new implicit numerical methods for the fractional cable equation[C]Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego, California, 2009.
[14]	Liu F, Yang Q, Turner I. Two new implicit numerical methods for the fractional cable equation[J]. Journal of Computational and Nonlinear Dynamics, 2011, 6(1): 011009.
[15]	Chen C, Liu F, Turner I, Anh V. Numerical methods with fourth-order spatial accuracy for variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid[J]. Computer & Mathematics With Application, 2011, 62(3): 971-986.
[16]	WU Chun-hong. Numerical solution for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative[J]. Appl Numer Math, 2009, 59(10): 2571-2583.
[17]	CUI Ming-rong. Compact finite difference method for the fractional diffusion equation[J]. J Comput Phys, 2009, 228(20): 7792-7804.
[18]	Zhuang P, Liu F, Anh V, Turner I. New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation[J]. Siam J Numer Anal, 2008, 46(2): 1079-1095.
[19]	Thomas J W. Numerical Partial Differential Equations: Finite Difference Methods[M]. New York: Springer-Verlag, 1995.
[20]	Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations, Springer Series in Compututational Mathematics 23[M]. Berlin: Springer-Verlag, 1994.
[21]	Roger A Horn, Charles R Johnson. Matrix Analysis[M]. Cambridge: Cambridge University Press, 1985.
[22]	叶超, 骆先南, 文立平. 分数阶扩散方程的一种新的高阶数值方法[J]. 湘潭大学自然科学学报, 2011, 33(4): 19-23.(YE Chao, LUO Xian-nan, WEN Li-ping. A new high order numerical method for the fractional diffusion equation[J]. Natural Science Journal of Xiangtan University, 2011, 33(4): 19-23.(in Chinese))