Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem

作者: Richard Sharp , Yen-Hsi Tsai , Björn Engquist

DOI: 10.1007/3-540-26444-2_13

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摘要: In this article, we study a class of numerical ODE schemes that use time filtering strategy and operate in two scales. The algorithms follow the framework heterogeneous multiscale methods (HMM) [1]. We apply to compute averaged path inverted pendulum under highly oscillatory vertical forcing on pivot. equation for related problems has been studied analytically [9]. prove show numerically proposed approximate thus average pendulum.

参考文章(10)
Bjorn Engquist, Yen-Hsi Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations Mathematics of Computation. ,vol. 74, pp. 1707- 1742 ,(2005) , 10.1090/S0025-5718-05-01745-X
B. García-Archilla, J. M. Sanz-Serna, R. D. Skeel, Long-Time-Step Methods for Oscillatory Differential Equations SIAM Journal on Scientific Computing. ,vol. 20, pp. 930- 963 ,(1999) , 10.1137/S1064827596313851
Weinan E, Bjorn Engquist, The Heterognous Multiscale Methods Communications in Mathematical Sciences. ,vol. 1, pp. 87- 132 ,(2003) , 10.4310/CMS.2003.V1.N1.A8
C. W. Gear, D. R. Wells, Multirate linear multistep methods BIT. ,vol. 24, pp. 484- 502 ,(1984) , 10.1007/BF01934907
Mark Levi, Geometry and physics of averaging with applications Physica D: Nonlinear Phenomena. ,vol. 132, pp. 150- 164 ,(1999) , 10.1016/S0167-2789(99)00022-6
Linda R. Petzold, An Efficient Numerical Method for Highly Oscillatory Ordinary Differential Equations SIAM Journal on Numerical Analysis. ,vol. 18, pp. 455- 479 ,(1981) , 10.1137/0718030
Ben Leimkuhler, Sebastian Reich, A reversible averaging integrator for multiple time-scale dynamics Journal of Computational Physics. ,vol. 171, pp. 95- 114 ,(2001) , 10.1006/JCPH.2001.6774
Xiantao Li, Weinan E, Multiscale modeling of the dynamics of solids at finite temperature Journal of The Mechanics and Physics of Solids. ,vol. 53, pp. 1650- 1685 ,(2005) , 10.1016/J.JMPS.2005.01.008
C. W. Gear, K. A. Gallivan, Automatic methods for highly oscillatory ordinary differential equations Lecture Notes in Mathematics. pp. 115- 124 ,(1982) , 10.1007/BFB0093152
Marlis Hochbruck, Arieh Iserles, Christian Lubich, Ernst Hairer, Geometric numerical integration ,(2006)