Phase errors and their correction in symplectic implicit single-step algorithm

For dynamic systems, an algorithm without theaccumulation of phase errors and magnitude errors isideal, but not easy to be found. Although there is no accumulation ofmagnitudes errors in Symplecticdifference algorithms, but the phase errors do accumulate. In this paper, thephase errors of single-step implicit symplectic algorithms, such assymplectic RK algorithms, are analyzed for linear systems withdamping and without damping, and the correction technique is proposed, which issimple and practical. The Euler implicit midpoint integration algorithm isadopted to solve several linear problems, and the results show that thecorrection technique of phase errors is valid. It is concludedthat the phase error is directly related to the quantity h\omega ,which is the product of time step and frequency. The higher the frequency,the higher the phase error; and it is hard to control the phaseerrors for MDOF systems.