Burgers方程的数值模拟方法
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摘要
本文主要讨论了在实际问题中遇到的两类偏微分方程的数值解法.主要研究了Burgers方程的隐-显多步有限元方法和分裂型最小二乘混合有限元格式,并且对一类反应扩散方程的隐-显多步有限元方法进行了研究.隐-显多步有限元方法结合了隐显式多步法和有限元方法,首先在空间层上利用有限元方法得到方程的弱形式,然后对时间变量运用隐显式多步法进行离散,从而得到稳定的、相容的、有效的演绎格式.这种格式能够得到时间方向的高精度离散,可以更广泛的应用于对时间精度要求较高的实际问题中.分裂型最小二乘混合有限元方法首先在时间层上将方程进行离散,引入未知变量后,便可得到一个耦合的方程组,然后进行区域分解,把方程组分成两个独立的子系统,这样就会使原问题的求解规模和难度得到很大程度的降低.通过理论上的分析,本文得到了该方法对原有未知变量具有L2 (Ω)最优阶误差估计.本文重点进行了隐-显多步有限元方法和分裂型最小二乘混合有限元方法的理论研究,研究结果表明本文所运用的两种数值模拟方法是可行的.
本文共分四章:
第一章绪论部分简要介绍了有限元方法的发展历程以及本文所用到的基本理论知识.
第二章研究了Burgers方程的隐-显多步有限元格式.
第三章研究了Burgers方程的分裂型最小二乘混合有限元格式.
第四章研究了一类非线性反应扩散方程的隐-显多步有限元格式.
This paper focuses on the numerical solutions to two partial differential equations. This paper studies implicit-explicit multistep finite element method and two splitting least squares mixed finite element method for Burgers equation, and use implicit-explicit multistep finite element method to study reaction-diffusion equations. Implicit-explicit multistep finite element method combines the implicit and explicit multistep method and finite element method, One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient. Two splitting least squares mixed finite element method discretizes the equations in time layer firstly, then introduce an unknown variable to get a coupled equations. And then decompose the domain, the coupled system can be splited into two independent sub-systems, so reduce the difficulty and scale of primal problems. Theoretical analysis shows that the method can get approximate solutions for the primal problems with optimal accuracy in ( )L2Ωnorm. This paper theoretical analysis on implicit-explicit multistep finite element method and two splitting least squares mixed finite element method, and verify the reasonableness of the two structures.
The main results of this paper are outlined as follows: In chapter one, the paper briefly introduces the development process of finite element method and some basic knowledge used in the latter part of this paper.
In chapter two, the paper studies implicit-explicit multistep finite element method for Burgers equation.
In chapter three, the paper studies two splitting least squares mixed finite element method for Burgers equation.
In chapter four, the paper studies implicit-explicit multistep finite element method for some nonlinear reaction-diffusion equations.
本文共分四章:
第一章绪论部分简要介绍了有限元方法的发展历程以及本文所用到的基本理论知识.
第二章研究了Burgers方程的隐-显多步有限元格式.
第三章研究了Burgers方程的分裂型最小二乘混合有限元格式.
第四章研究了一类非线性反应扩散方程的隐-显多步有限元格式.
This paper focuses on the numerical solutions to two partial differential equations. This paper studies implicit-explicit multistep finite element method and two splitting least squares mixed finite element method for Burgers equation, and use implicit-explicit multistep finite element method to study reaction-diffusion equations. Implicit-explicit multistep finite element method combines the implicit and explicit multistep method and finite element method, One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient. Two splitting least squares mixed finite element method discretizes the equations in time layer firstly, then introduce an unknown variable to get a coupled equations. And then decompose the domain, the coupled system can be splited into two independent sub-systems, so reduce the difficulty and scale of primal problems. Theoretical analysis shows that the method can get approximate solutions for the primal problems with optimal accuracy in ( )L2Ωnorm. This paper theoretical analysis on implicit-explicit multistep finite element method and two splitting least squares mixed finite element method, and verify the reasonableness of the two structures.
The main results of this paper are outlined as follows: In chapter one, the paper briefly introduces the development process of finite element method and some basic knowledge used in the latter part of this paper.
In chapter two, the paper studies implicit-explicit multistep finite element method for Burgers equation.
In chapter three, the paper studies two splitting least squares mixed finite element method for Burgers equation.
In chapter four, the paper studies implicit-explicit multistep finite element method for some nonlinear reaction-diffusion equations.
引文
[1] J.R.Ockendon, S.D.Howison, A.A.Lacey, et.al, Applied Partial Differential Equations. London: Oxford University Press, 1999.
[2] Burgers J.M., A mathematical model illustrating the theory of turbulence. New York: Adv in Appl Mech I, Academic Press, 1948.
[3] Evans.D.J, Sahimi.M.S., The numerical solution of Burgers equations by the alternation group explicit (AGE) method. Inter J Computer Math., 1989, 29:39-64 .
[4]孙海燕,谢树森,一维Burgers方程的一类交替分段并行算法,中国海洋大学学报, 2006, 36:215-218.
[5] Gu Haiming, Li Hongwei, An adaptive least-squares mixed finite element method for Burgers equations, Computational Mathematics and Mathematical Physics, 2009, 49(4):647-656.
[6] Jiang C S, Wang M X, Xie C H.Asymptotic Behavior of Global Sulutions for a Chemical Rcaction Model[J].J Math Anal Appl, 1998, 222(2):640-656.
[7] Jiang C S, Xie C H. Asymptotic Behavior of the Sulutions to Chemical [J].Chinese J of Contemporary Math, 1997, 18(2):137-149.
[8] Wang M X. Asymptotic Behavior of the Sulutions to Some Rcaction-diffusion Systom[J]. Chinese J of Contemporary Math, 1997, 18(3):124-136.
[9]江成顺,崔霞.一类非线性反应扩散方程组的有限元分析[J].计算数学. 2000.2.22(1): 103-112.
[10]李杰,几类偏微分方程的混合有限元方法, [学位论文]:青岛科技大学, 2010.
[11]冯康,基于变分原理的差分格式,应用数学与计算数学, 1965, 2(4):238-262.
[12] Georgios Akrivis, Michel Crouzeix, Charalambos Makridakis, Implicit-Explicit multistp finite element methods for nonlinear parabolic problems.Math of Comp, 1998, 67:457-477.
[13] Babuska I, Error-bounds for finite element method, Numer.math., 1971,16:322-333.
[14] Brezzi F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, SIAM J.Numer.Anal., 1974, 13:185-197.
[15] J.H.Bramble and J.A.Nitsche, A generalized Ritz-least-squares method for Dirichlet problems, SIAM J.Numer.Anal, 1973, 10:81-93.
[16] V.Girault and P.A.Raviart, Finite element method for Navier-Stokes equations, Theory and Algorithms Springer, Berlin, 1980.
[17] G.F.Carey and J.T.Oden, Finite Elements, Prentice-Hall, Engle-wood Cliffs, 1983, 11:1289-1306.
[18] Gu Haiming, Yang Dangping, Sui Shulin, et.al, Least-squares Mixed Finite Element Method for A Class of Stokes Equation, Appl.Math.and Mech., 2000, 21(5):557-566.
[19] Gu Haiming, Xu Xiuling, The Least-squares Mixed Finite Element Methods for a Degenerate Elliptic Problem, Math.Appl., 2002, 15(1):118-122.
[20] Gu Haiming and Yang Danping, Least-Squares Mixed Finite Element Method for a class of Stokes equation, Applied Mathematics and Mechanics, 2000, 21:501-510.
[21] Gu Haiming, Li Hongwei, An adaptive least-squares mixed finite element method for nonlinear parabolic problems, Computational Mathematics and Modeling, 2009, 20(2):192-206.
[22] Gu Haiming and Yang Danping, Least-Squares Mixed Finite Element Method for sobolev equations, Indian J.Pure Appl.Math., 2000, 31(5):505-517.
[23] Luo Zhendong and Zhu Jiang, A Nonlinear Galerkin/Petrov Least-Squares Mixed Element Method for the Stationary Navier-Stokes Equations, Applied Mathematics and Mechanics, 2002, 23:697-706.
[24] Chen Y.P., Huang Y.Q. and Shen Z.H., Least-Squares Mixed Finite Element Method for degenerate elliptic problems, Numer.Math.Sinica, 2001, 23:87-94.
[25] Chen Y.P., Huang Y.Q. and Shen Z.H., The Least-Squares Mixed Finite Element Method for incompressible Miscible displacements, Mathematical Numerical Snica, 2000, 3: 270-278.
[26] Chen Y. and Huang Y.Q., Superconvergence of least-squares mixed finite element for symmetric elliptic problems, Numer.Math.Sinica, 1999, 21:80-94.
[27] Chen Y. and Huang Y.Q., The full-discrete mixed finite element methods for non- linear hyperbolic equations, Numer.Math.Sinica, 2000, 22:173-194.
[28] Chen Y. and Huang Y.Q., The superconvergence of mixed finite element methods for nonlinear hyperbolic equations, Numer.Math.Sinica, 1997, 16:87-94.
[29]鲁祖亮, OldroydB型流体流动的混合有限元方法和收敛性质的研究, [学位论文]:长沙理工大学, 2007.
[30]罗振东,混合有限元法基础及其应用,北京:科学出版社, 2006.
[31] Michel Crouzeix, An Implicit-Explicit multistp method of the approximation of parabolic equations.Numer Math, 1980, 35:257-276.
[32] Lamber J D.Computation methods in ordinary differential equations.New York:John Wiley Son, 1973.
[33] Georgios Akrivis, Michel Crouzeix, Charalambos Makridakis, Implicit-Explicit multistp methods for quasilinear parabolic problems. .Numer Math, 1999, 82:521-541.
[34] Thome V.Galerkin finite element methods for parabolic problems.Lecture Notes in Mathmatics 1054, Berlin:Spinger-Verlag, 1984.
[35] Georgios Akrivis Fotini Karakatsani. Modified Implicit–Explicit BDF Methods for Nonlinear Parabolic Equations. BIT Numerical Mathematics, Volume 43, Number 3: 467-483.
[36]陈蔚.一类反应扩散方程组的隐-显多步有限元方法.数学物理学报, 2002, 22A(2):180-188.
[37] Chen Wei. Implicit-Explicit Multistep Finite Element Methods for A Semiconductor Device With Heat Conduction. Journal of Mathematical Study, 2002(02)
[38] Ciarlet P.G. The Finite Element Method for Elliptic Problems[M].Amsterdam: North-Holland.1978:110—168.
[39] Evans D J, Sahimi M S. The numerical solution of Burgers equations by the alternation group explicit (AGE) method [J]. Inter J Computer Math , 1989, 29:39-64.
[40]高夫征,芮洪兴.求解线性Sobolev方程的分裂型最小二乘混合元方法[J].计算数学.2008, 30(3):269—282.
[41] Haiming Gu, Characteristic Finite Element Methods for Nonlinear Sobolev Equations[J].1999, 102:51—62.
[42] Gu Haiming, Yang Danping, Sui Shulin, Liu Xinmin. Least-squares Mixed Finite Element Method for a Class of Stokes Equation[J]. Applied Mathematics and Mechanics.2000, 21(5):557—566.
[43] Haiming Gu, Xiaonan Wu. Alternating-direction Iterative Technique for Mixed Finite Element Methods for Stokes Equations[J]. Applied Mathematics and Computation.2005, 162:1035—1047.
[44] J M Burgers. A Mathematical Model Illustrating the Theory of Turbulence. [M]. New York: Adv in Appl Mech I, Academic Press, 1948:171-199.
[45] Zhiqiang Cai, Johannes Korsawe, Gerhard Starke. An Adaptive Least Squares Mixed Finite Element Method for the Stress-Displacement Formulation of Linear Elasticity[J]. Numer Methods Partial Differential Eq. 2005, 21:132-148 .
[46] Oden J T, Babuska, Baumann C E.A discontinuous h-p finite element method for diffusion problems[J].J Comput Phys, 1998, 14:495-519.
[47] Biswas R, Devine K D, Flaherty J E. Parallel, adaptive finite element method for conservation laws[J]. Applied Numer Math, 1994, 14:255-283.
[2] Burgers J.M., A mathematical model illustrating the theory of turbulence. New York: Adv in Appl Mech I, Academic Press, 1948.
[3] Evans.D.J, Sahimi.M.S., The numerical solution of Burgers equations by the alternation group explicit (AGE) method. Inter J Computer Math., 1989, 29:39-64 .
[4]孙海燕,谢树森,一维Burgers方程的一类交替分段并行算法,中国海洋大学学报, 2006, 36:215-218.
[5] Gu Haiming, Li Hongwei, An adaptive least-squares mixed finite element method for Burgers equations, Computational Mathematics and Mathematical Physics, 2009, 49(4):647-656.
[6] Jiang C S, Wang M X, Xie C H.Asymptotic Behavior of Global Sulutions for a Chemical Rcaction Model[J].J Math Anal Appl, 1998, 222(2):640-656.
[7] Jiang C S, Xie C H. Asymptotic Behavior of the Sulutions to Chemical [J].Chinese J of Contemporary Math, 1997, 18(2):137-149.
[8] Wang M X. Asymptotic Behavior of the Sulutions to Some Rcaction-diffusion Systom[J]. Chinese J of Contemporary Math, 1997, 18(3):124-136.
[9]江成顺,崔霞.一类非线性反应扩散方程组的有限元分析[J].计算数学. 2000.2.22(1): 103-112.
[10]李杰,几类偏微分方程的混合有限元方法, [学位论文]:青岛科技大学, 2010.
[11]冯康,基于变分原理的差分格式,应用数学与计算数学, 1965, 2(4):238-262.
[12] Georgios Akrivis, Michel Crouzeix, Charalambos Makridakis, Implicit-Explicit multistp finite element methods for nonlinear parabolic problems.Math of Comp, 1998, 67:457-477.
[13] Babuska I, Error-bounds for finite element method, Numer.math., 1971,16:322-333.
[14] Brezzi F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, SIAM J.Numer.Anal., 1974, 13:185-197.
[15] J.H.Bramble and J.A.Nitsche, A generalized Ritz-least-squares method for Dirichlet problems, SIAM J.Numer.Anal, 1973, 10:81-93.
[16] V.Girault and P.A.Raviart, Finite element method for Navier-Stokes equations, Theory and Algorithms Springer, Berlin, 1980.
[17] G.F.Carey and J.T.Oden, Finite Elements, Prentice-Hall, Engle-wood Cliffs, 1983, 11:1289-1306.
[18] Gu Haiming, Yang Dangping, Sui Shulin, et.al, Least-squares Mixed Finite Element Method for A Class of Stokes Equation, Appl.Math.and Mech., 2000, 21(5):557-566.
[19] Gu Haiming, Xu Xiuling, The Least-squares Mixed Finite Element Methods for a Degenerate Elliptic Problem, Math.Appl., 2002, 15(1):118-122.
[20] Gu Haiming and Yang Danping, Least-Squares Mixed Finite Element Method for a class of Stokes equation, Applied Mathematics and Mechanics, 2000, 21:501-510.
[21] Gu Haiming, Li Hongwei, An adaptive least-squares mixed finite element method for nonlinear parabolic problems, Computational Mathematics and Modeling, 2009, 20(2):192-206.
[22] Gu Haiming and Yang Danping, Least-Squares Mixed Finite Element Method for sobolev equations, Indian J.Pure Appl.Math., 2000, 31(5):505-517.
[23] Luo Zhendong and Zhu Jiang, A Nonlinear Galerkin/Petrov Least-Squares Mixed Element Method for the Stationary Navier-Stokes Equations, Applied Mathematics and Mechanics, 2002, 23:697-706.
[24] Chen Y.P., Huang Y.Q. and Shen Z.H., Least-Squares Mixed Finite Element Method for degenerate elliptic problems, Numer.Math.Sinica, 2001, 23:87-94.
[25] Chen Y.P., Huang Y.Q. and Shen Z.H., The Least-Squares Mixed Finite Element Method for incompressible Miscible displacements, Mathematical Numerical Snica, 2000, 3: 270-278.
[26] Chen Y. and Huang Y.Q., Superconvergence of least-squares mixed finite element for symmetric elliptic problems, Numer.Math.Sinica, 1999, 21:80-94.
[27] Chen Y. and Huang Y.Q., The full-discrete mixed finite element methods for non- linear hyperbolic equations, Numer.Math.Sinica, 2000, 22:173-194.
[28] Chen Y. and Huang Y.Q., The superconvergence of mixed finite element methods for nonlinear hyperbolic equations, Numer.Math.Sinica, 1997, 16:87-94.
[29]鲁祖亮, OldroydB型流体流动的混合有限元方法和收敛性质的研究, [学位论文]:长沙理工大学, 2007.
[30]罗振东,混合有限元法基础及其应用,北京:科学出版社, 2006.
[31] Michel Crouzeix, An Implicit-Explicit multistp method of the approximation of parabolic equations.Numer Math, 1980, 35:257-276.
[32] Lamber J D.Computation methods in ordinary differential equations.New York:John Wiley Son, 1973.
[33] Georgios Akrivis, Michel Crouzeix, Charalambos Makridakis, Implicit-Explicit multistp methods for quasilinear parabolic problems. .Numer Math, 1999, 82:521-541.
[34] Thome V.Galerkin finite element methods for parabolic problems.Lecture Notes in Mathmatics 1054, Berlin:Spinger-Verlag, 1984.
[35] Georgios Akrivis Fotini Karakatsani. Modified Implicit–Explicit BDF Methods for Nonlinear Parabolic Equations. BIT Numerical Mathematics, Volume 43, Number 3: 467-483.
[36]陈蔚.一类反应扩散方程组的隐-显多步有限元方法.数学物理学报, 2002, 22A(2):180-188.
[37] Chen Wei. Implicit-Explicit Multistep Finite Element Methods for A Semiconductor Device With Heat Conduction. Journal of Mathematical Study, 2002(02)
[38] Ciarlet P.G. The Finite Element Method for Elliptic Problems[M].Amsterdam: North-Holland.1978:110—168.
[39] Evans D J, Sahimi M S. The numerical solution of Burgers equations by the alternation group explicit (AGE) method [J]. Inter J Computer Math , 1989, 29:39-64.
[40]高夫征,芮洪兴.求解线性Sobolev方程的分裂型最小二乘混合元方法[J].计算数学.2008, 30(3):269—282.
[41] Haiming Gu, Characteristic Finite Element Methods for Nonlinear Sobolev Equations[J].1999, 102:51—62.
[42] Gu Haiming, Yang Danping, Sui Shulin, Liu Xinmin. Least-squares Mixed Finite Element Method for a Class of Stokes Equation[J]. Applied Mathematics and Mechanics.2000, 21(5):557—566.
[43] Haiming Gu, Xiaonan Wu. Alternating-direction Iterative Technique for Mixed Finite Element Methods for Stokes Equations[J]. Applied Mathematics and Computation.2005, 162:1035—1047.
[44] J M Burgers. A Mathematical Model Illustrating the Theory of Turbulence. [M]. New York: Adv in Appl Mech I, Academic Press, 1948:171-199.
[45] Zhiqiang Cai, Johannes Korsawe, Gerhard Starke. An Adaptive Least Squares Mixed Finite Element Method for the Stress-Displacement Formulation of Linear Elasticity[J]. Numer Methods Partial Differential Eq. 2005, 21:132-148 .
[46] Oden J T, Babuska, Baumann C E.A discontinuous h-p finite element method for diffusion problems[J].J Comput Phys, 1998, 14:495-519.
[47] Biswas R, Devine K D, Flaherty J E. Parallel, adaptive finite element method for conservation laws[J]. Applied Numer Math, 1994, 14:255-283.