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Kuramoto-Sivashinsky方程是一种可以描述复杂混沌现象的高阶非线性演化方程. 方程中高阶导数项的存在,使得传统无单元Galerkin方法采用高次多项式基函数构造形函数时, 形函数违背了一致性条件.因此,本文提出了一种采用平移多项式基函数的无单元Galerkin方法. 与传统无单元Galerkin方法相比,该方法在方程离散时依然采用Galerkin进行离散, 但形函数的构造采用了基于平移多项式基函数的移动最小二乘近似. 通过对具有行波解和混沌现象的Kuramoto-Sivashinsky方程的数值模拟,验证了本文方法的有效性.
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关键词:
- 无单元Galerkin方法 /
- Kuramoto-Sivashinsky方程 /
- 平移多项式基函数 /
- 混沌现象
The Kuramoto-Sivashinsky equation is a kind of high-order nonlinear evolution equation which can describe complicated chaotic nature. Due to the existence of high-order derivatives in the equation, the shape functions violate the consistency conditions when using traditional element-free Galerkin method which adopts high-order polynomial basis functions to construct the shape functions. In order to solve the problems encountered in the traditional element-free Galerkin method, a kind of element-free Galerkin method adopting the shifted polynomial basis functions is presented in this paper. Compared with the traditional element-free Galerkin method, the Galerkin principle is still used to discrete the equation in this method, but the shape functions are constructed by moving least squares based on the shifted polynomial basis functions. Numerical results for the Kuramoto-Sivashinsky equation having traveling wave solution and chaotic nature prove the validity of the presented method.-
Keywords:
- element-free Galerkin /
- Kuramoto-Sivashinsky equation /
- shifted polynomial basis functions /
- chaotic nature
[1] Kuramoto Y, Tsuzuki T 1975 Prog. Theor. Phys. 54 687
[2] Sivashinsky G I 1977 Acta Astrorsautica 4 1177
[3] Hyman J M, Nicolaenko B 1986 Physica D 18 113
[4] Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) p120 (in Chinese) [刘秉正, 彭建华 2004 非线性动力学 (北京:高等教育出版社) 第120页]
[5] Kuramoto Y, Tsuzuki T 1976 Pron. Theor. Phys. 55 356
[6] Sivashinsky G I, Michelson D M 1980 Pron. Theor. Phys. 63 2112
[7] Sivashinsky G I 1983 Ann. Rev. Fluid Mech. 15 179
[8] Fan E 2000 Phys. Lett. A 277 212
[9] Peng Y Z 2003 Commun. Theor. Phys. 39 641
[10] Nickel J 2007 Chaos, Solitons and Fractals 33 1376
[11] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[12] Cheng R J, Cheng Y M 2011 Chin. Phys. B 20 070206
[13] Jin X Z, Li G, Aluru N R 2001 Comput. Modell. Eng. Sci. 2 447
[14] Fernanez-Mendez S, Huerta A 2004 Comput. Methods Appl. Mech. Eng. 193 1257
[15] Uddin M, Haq S, Islam S U 2009 Appl. Math. Comput. 212 458
[16] Abdel-Gawad H I, Abdusalam H A 2001 Chaos, Solitons and Fractals 12 2039
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[1] Kuramoto Y, Tsuzuki T 1975 Prog. Theor. Phys. 54 687
[2] Sivashinsky G I 1977 Acta Astrorsautica 4 1177
[3] Hyman J M, Nicolaenko B 1986 Physica D 18 113
[4] Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) p120 (in Chinese) [刘秉正, 彭建华 2004 非线性动力学 (北京:高等教育出版社) 第120页]
[5] Kuramoto Y, Tsuzuki T 1976 Pron. Theor. Phys. 55 356
[6] Sivashinsky G I, Michelson D M 1980 Pron. Theor. Phys. 63 2112
[7] Sivashinsky G I 1983 Ann. Rev. Fluid Mech. 15 179
[8] Fan E 2000 Phys. Lett. A 277 212
[9] Peng Y Z 2003 Commun. Theor. Phys. 39 641
[10] Nickel J 2007 Chaos, Solitons and Fractals 33 1376
[11] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[12] Cheng R J, Cheng Y M 2011 Chin. Phys. B 20 070206
[13] Jin X Z, Li G, Aluru N R 2001 Comput. Modell. Eng. Sci. 2 447
[14] Fernanez-Mendez S, Huerta A 2004 Comput. Methods Appl. Mech. Eng. 193 1257
[15] Uddin M, Haq S, Islam S U 2009 Appl. Math. Comput. 212 458
[16] Abdel-Gawad H I, Abdusalam H A 2001 Chaos, Solitons and Fractals 12 2039
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