Symplectic And Exponentially Fitted Runge-Kutta (-Nystr(o|¨)m) Type Methods For Oscillatory Ordinary Differential Equations

A large number of problems arising in physical fields such as electric circuits, elastics, celestial mechanics, quantum physics and so on, are governed by systems of first or second order ordinary differential equations (ODEs). The focus of this thesis is on structure-presrerving numerical integrators of Runge-Kutta(-Nystrom) type for this category of initial value problems (IVPs) of ordinary differential equations.It has been more than one hundred years since the Runge-Kutta methods were initiated. The Runge-Kutta methods are mainly applied to first or second order initial value problems of ODEs. A system of second order ODEs can of course be turned into a system of first order ODEs by introducing a component of velocity and can be solved with a RK method. Afterwards Runge-Kutta-Nystrom methods were designed by Nystrom to solve second order problems in a direct way. On the other hand, in1960’s J. Butcher set up theories of rooted trees and B-series so that obtaining higher order RK methods became a reality. Recent research in applied sciences and engineering requires further that numerical algorithms preserve some significant properties of the original systems such as invariants of the systems, especially symplecticness and the Hamiltonian functions of the Hamiltonian systems of interest. We will concentrate on the structural features of oscillatory differential equations and on formulating effective numerical methods to preserve the qualitative behavior of the exact solutions.This thesis is divided into four chapters.Chapter1presents briefly the basic concepts relating to ordinary differential equations, including the existence and uniqueness of the solutions to initial value problems of ODEs, consistency, convergence and stability of numerical integrators. We also introduce symplectic Runge-Kutta methods solving Hamiltonian system and the exponential fitting technique, which in fact is one of the most frequently adopted structure-preserving procedures for oscillatory problems. Chapter2proposes a new family of integrators of RK type which we denote as RKNd methods. One of the advantages of the new methods is that their internal stages approximate the real solution in at least one order higher than those of the traditional Runge-Kutta methods (which are only of order one). Under suitable simplifying conditions we derive via order conditions three explicit and four implicit RKNd methods of two stages. Linear stability, the orders of dispersion and dissipation for the new methods are also examined. From the numerical results produced by these methods when applied to some typical first and second order problems, one finds that our new methods are more effective for the second order ODEs than first order problems.Chapter3is concerned with the exponentially fitted methods based on the new methods given in Chapter2. Order conditions for these exponentially fitted methods are obtained and three two-stage exponentially fitted RKNd (EFRKNd) methods of orders two, three and four, respectively, are constructed. The stability regions, the orders of dispersion and dissipation for these new methods are analyzed. The results of numerical experiments on several typical second order oscillatory problems show that the EFRKNd methods work more efficiently than the exponentially fitted RK methods of the same order.Chapter4focuses on the symmetric and symplectic exponentially fitted RKN methods solving Hamiltonian problems. We give the conditions of symmetry, symplecticness, exponential fitting and consistency for modified RKN methods under which we construct a two-stage method of order two, a three-stage method of order two and a four-stage method of order four. Also we check the periodicity regions, the orders of dispersion and dissipation of the new methods. The results of numerical experiments accompanied show that the new symmetric RKN methods obtained in this chapter integrate oscillators more efficiently and preserve the Hamiltonian function more exactly than non-symmetric methods.In summary, this thesis has made the following contributions:●a new family methods of RK type (RKNd) which are at least one more exact in the internal stages than traditional RK methods;●the exponentially fitted methods based on these new RKNd methods;●a family of symmetric, symplectic, and exponentially fitted RKN methods for Hamiltonian systems.