The Study Of Numerical Methods For Second-order Oscillatory Differential Equations

Oscillatory phenomena are frequently encountered in pure and applied mathemat-ics and in applied sciences such as mechanics, physics, astronomy, molecular biology and engineering. A lot of theoretical and numerical researches have been done on the modeling and simulation of these oscillations. Among typical topics is the numerical integration of second-order oscillatory system where M∈Rdxd is a symmetric and positive semi-definite matrix that implicitly con-tains the frequencies of the problem. A large number of excellent methods have been proposed and applied. However, most of the existed methods do not take into account the special structure of oscillatory system (2), so they are not satisfactory in applica-tions. The purpose of this work is to design and study the efficient methods adapted to the special structure of (2).The outline of this dissertation is as follows.Chapter1introduces the theory of rooted trees (Nystrom-trees) and B-series (Nystrom-series), by which the order conditions for classical Runge-Kutta (-Nystrom) methods are derived. The theory of B-series is the backbone of this dissertation.Chapter2generalizes Franco's ARKN methods from one-dimensional perturbed oscillators to systems. For one-dimensional perturbed oscillators, Franco [22] proposed ARKN methods and derived the order conditions based on the theory of Nystrom-trees. However, some critical mistakes have been made in the order conditions and their derivation in that paper. We derive the correct order conditions for ARKN methods via adapted Nystrom-series defined in this dissertation.Chapter3proposes and studies a family of two-step extended Runge-Kutta-Nystrom-type (TSERKN) methods which inherit the framework of two-step hybrid methods [13] and make full use of the special feature of the true flows for both the inter-nal stages and the updates. Note that TSERKN methods exactly integrate the equation y"+My=0. When M→0, these methods reduce to two-step hybrid methods. Order conditions for the new methods are derived via the BBT-series defined on the set BT of branches and the BBWT-series defined on the subset BWT of BT. Three practical TSERKN integrators are constructed. The results of the numerical experiments show that these integrators are more efficient than some codes in the recent literature.Chapter4constructs a new TSERKN method and shows that the global error bound for the new method is independent of‖M‖.Chapter5formulates the adapted Falkner methods which are an extension of the reformed Falkner methods proposed in [42] for the oscillatory problem (2) in the case when the right-side function does not contain y'. Like TSERKN methods, these meth-ods also exactly integrate the equation y"+My=0. As an important result, when a k-step adapted Falkner method is applied to (2), the global error bound of the solution is shown to be independent of‖M‖. The results of the numerical experiments show that the adapted Falkner methods are more efficient than the reformed Falkner methods.