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By Brian Osserman

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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.

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