By Xinyuan Wu, Kai Liu, Wei Shi
This e-book describes a number of powerful and effective structure-preserving algorithms for second-order oscillatory differential equations. Such platforms come up in lots of branches of technological know-how and engineering, and the examples within the e-book comprise platforms from quantum physics, celestial mechanics and electronics. To safely simulate the real habit of such structures, a numerical set of rules needs to shield up to attainable their key structural houses: time-reversibility, oscillation, symplecticity, and effort and momentum conservation. The e-book describes novel advances in RKN tools, ERKN tools, Filon-type asymptotic tools, AVF tools, and trigonometric Fourier collocation equipment. The accuracy and potency of every of those algorithms are validated through cautious numerical simulations, and their structure-preserving homes are conscientiously verified through theoretical research. The e-book additionally offers insights into the sensible implementation of the methods.
This publication is meant for engineers and scientists investigating oscillatory platforms, in addition to for lecturers and scholars who're attracted to structure-preserving algorithms for differential equations.
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This e-book describes quite a few powerful and effective structure-preserving algorithms for second-order oscillatory differential equations. Such structures come up in lots of branches of technological know-how and engineering, and the examples within the ebook contain structures from quantum physics, celestial mechanics and electronics.
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Extra info for Structure-Preserving Algorithms for Oscillatory Differential Equations II
For this Hamiltonian problem, long time-step methods may lead to the problem of numerically induced resonance instabilities. 02). We consider the long-time interval [0, 1000]. 11 present the results of the different methods. Method SV shows a poor energy conservation when hω is more than about π/2, while methods ISV1 and ISV2 do poorly only near integral multiples of π . Method A shows poor energy conservation near even multiples of π and only the method B shows a uniformly good behaviour for all frequencies.
1007/s10208-014-9241-9 19. Wu X (2014) Matrix-variation-of-constants formula with applications. A seminar report of Nanjing University (preprint) 20. Wu X, Mei L, Liu C (2015) An analytical expression of solutions to nonlinear wave equations in higher dimensions with Robin boundary conditions. J Math Anal Appl 426:1164–1173 21. Wu X, You X, Shi W, Wang B (2010) ERKN integrators for systems of oscillatory second-order differential equations. Comput Phys Commun 181:1873–1887 22. Wu X, You X, Xia J (2009) Order conditions for ARKN methods solving oscillatory systems.
Method SV shows a poor energy conservation when hω is more than about π/2, while methods ISV1 and ISV2 do poorly only near integral multiples of π . Method A shows poor energy conservation near even multiples of π and only the method B shows a uniformly good behaviour for all frequencies. 02 0 π 0 2π 3π π 4π 2π 3π 4π hω hω Fig. 5. 2 π 0 2π hω 3π 4π 0 π 2π hω 3π 4π π 2π hω 3π 4π Fig. 5. 4) and the two Gautschi-type methods A and B. It is noted that the formula ISV2 behaves better than the formula ISV1.