Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type

Abstract

Abstract. The convergence behaviour of multi-revolution composition methods combined with timesplitting methods is analysed for highly oscillatory linear differential equations of Schrödinger type. Numerical experiments illustrate and complement the theoretical investigations.Mathematics Subject Classification. 34K33, 34G10, 35Q41, 65M12, 65N15.

Simply improved averaging for coupled oscillators and weakly nonlinear waves

“…Nevertheless, it is necessary to mention great existing works in the active field of highly oscillatory system simulations, and the list by no means can be complete. For instance, (i) since the formulation of envelope following method [85], clever approaches that address one high-frequency (which, in the case of eq.1, corresponds to Ω with eigenvalues all being integer multiples of one value) have been continuously constructed, such as multi-revolution composition method [29,30], stroboscopic averaging method (which can be made high-order in ǫ) [23,22,26], the two-scaled reformulation (which is uniformly accurate after adding one artificial dimension) [27], and additional uniformly accurate approaches based on various formal asymptotic expansions [12,11,13,35]. Tools for performance analysis have been proposed too, such as modulated Fourier expansion [50,32], which in turn accelerates the development of numerical methods, for instance, for second-order differential equations (see e.g., a seminal work [31], a specific investigation [92], and a recent work [114] in which 2nd-order uniform accuracy was achieved on half of the variables for polynomial potentials).…”

“…Remarks on dynamics of the averaged system (30). At least for the parameters used in [96], the fixed point identified above can be verified by linearization to be asymptotic stable (i.e., a sink), and hence it corresponds to a steady state of the CPUT.…”