Dongbing  Liu.  Study on oscillatory ordinary differential equation based on linear multistep algorithm.  Dynamic Systems and Applications 29 (2020) No. 3, 522-541

https://doi.org/10.46719/dsa202029310

ABSTRACT.
Periodic oscillatory differential equation is widely used in biology, chemistry, astronomy and other fields. The solution of differential equation is difficult to calculate, so the numerical solution is very important. The commonly used numerical integration methods include Runge Kutta (RK) method, Runge Kutta Nystrom (RKN) method and linear multi-step method (LMM). The linear multi-step method has the characteristics of high accuracy, high efficiency and simple structure. The purpose of this paper is to study the structure preserving linear multistep method for solving the first and second order oscillation problems, and to solve the oscillation differential equation by the linear multistep method. Inspired by the solution structure of linear harmonic oscillator, a new P-STABLE symmetric extended linear multi-step method (Selm) is established, and its corresponding prediction and correction are carried out. Based on the principle that the coefficient depends on P-stability, two-step explicit and implicit Selm methods, four step explicit and implicit Selm methods, six step explicit and implicit Selm methods and eight step explicit and implicit Selm methods are constructed. A new prediction correction eight step selmpc (selmpc) method is obtained by using the explicit method and the implicit Selm method. The results show that the new selmpc method has the smallest error coefficient and is significantly better than the synchronous selmpc method in other literatures.

Keywords.  linear multistep algorithm; oscillatory differential equation; order condition; fitting