Linear Multistep Methods For Oscillatory Ordinary Differential Equations

Periodic or oscillatory differential equations arise naturally in a variety of applied sci-ences such as molecular dynamics, astronomy, biological chemistry, quantum mechanics elec-tronics and bioengineering. The analytic solutions to most periodic or oscillatory differential equations are difficult to obtain, high accuracy of integration of the equation have become extraordinary important. Until now there have been broadly three categories of numerical integrators for solving initial value problems of ordinary differential equations:Runge-Kutta (RK) methods, Runge-Kutta-Nystrom (RKN) methods and linear multistep (LM) methods. Compared with RK and RKN methods, the greatest advantage of multistep methods lie in simplicity, high accuracy and efficiency. The past decade has seen a vast of work in RK and RKN methods adapted to integration of oscillatory differential equations. This thesis aims at developing highly effective structure-preserving linear multistep methods for first-order and second-order oscillatory problems.The thesis is divided into four chapters.Chapter 1, as preliminaries of this thesis, surveys some important properties of linear multistep methods for initial value problems of first-order differential equations, focusing on order conditions and convergence. It is shown that consistent linear multistep methods preserve linear invariants. For symmetric linear multisteps methods solving second-order ordinary differential equations, a new concept of pseudo-phase lag is proposed, based on which a new equivalent condition for exponential fitting is set up via the derivatives of the pseudo-phase lag. For linear multistep methods solving higher order ordinary differential equations, an equivalent condition of algebraic order is presented and shown.In Chapter 2, we construct a family of explicit eight-step methods with vanished pseudo-phase lag, derivatives and integrals of pseudo-phase lag. Numerical results are presented from applying the new method to two well-known periodic orbital problems. The numerical results show that the new method is more efficient than Simos’s method proposed in 2004 for several choices of fitting frequencies.In Chapter 3, we explore properties two derivatives linear multistep (TDLM) methods. A formula of local error is given in terms of linear operators. A necessary condition for convergence is shown. It is also shown that consistent two-derivative linear mulitstep methods preserve linear invariants. An equivalent condition for algebraic order is presented and shown.In Chapter 4, motivated by a special structure of the solution of the linear harmonic oscillator, we propose a new central difference operator, based on which we build up a novel family of P-stable symmetric extended linear multistep methods. Specific practical explicit and implicit two-step, four-step, six-step and eight-step SELM method are constructed. A predictor-corrector method (SELMPC) is also obtained by embedding the eight-step explicit SELM predictor and the eight-step implicit SELM corrector. Error analysis is carried out for every new method. The error the coefficient of the new predictor-corrector method is shown to the minimum compared with some integrators of the same order from the literature. Numerical results show that the new methods are more effective than some existing symmetric linear methods with the same step which have the same algebraic order.The last part surveys the main contributions of this thesis, and some challenging topics are put forward for future work.