报告题目：Efficient oscillation-preserving  integrators for multiple high-frequency  oscillatory second-order ODEs: a review with analysis I, II, III

报告人：吴新元  2022白菜注册网站特聘教授、南京大学教授

时    间：I: 2019年3月25号 下午 16:00-17:00

     II: 2019年4月29号 下午 16:00-17:00

        III: 2019年5月9号 下午 16:00-17:00

地    点：002cc白菜资讯三楼报告厅304

主办单位：002cc白菜资讯

报告摘要:

In the last few decades, Runge-Kutta-Nystr\"om (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this talk, from the point of view of Geometric Integration, this progress is reviewed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. We then analyse the oscillation-preserving behaviour of RKN-type methods, including extended RKN (ERKN) integrators, adapted RKN (ARKN) methods, Trigonometric Fourier collocation (TFC) methods, Average-Vector-Field (AVF) methods, adapted Average-Vector-Field (AAVF) methods, symplectic and symmetric RKN methods, and standard RKN methods for solving the nonlinear multi-frequency highly oscillatory system of second-order differential equations y''+My=f(y,y') with initial values y(0)=y_0, y'(0)=y_0', where M is a positive semi-definite matrix. It turns out that both the internal stages and the updates of ERKN integrators and TFC methods respect the qualitative and global features of the highly oscillatory solution, in the light of the matrix-variation-of-constants formula associated with the nonlinear multi-frequency highly oscillatory system, whereas the internal stages of both RKN and ARKN methods suffer from a fatal weakness which damages the oscillation preservation. Hence, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for a numerical method within the broader framework on the subject of the geometric numerical integration, and the remarkable superiority of oscillation-preserving integrators for solving nonlinear multi-frequency highly oscillatory systems.

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