Splitting Methods For Oscillatory Differential Equations With Applications To Biology

Oscillatory differential equations play an important role in a wide variety of applied fields in science and engineering,such as molecular dynamics,astronomy,biology and so on.However,it is difficult in general to obtain the analytic solution to a differential equation.Therefore,effective numerical solution will be the significant way to investigate the behaviors of oscillatory differential equations.Splitting methods form one of the most important and powerful tools for numerical investigation of complicated systems which can be decomposed into some subsystems in an appropriate way.Despite the simple forms,splitting methods,usually explicit,easy to be implemented,can preserve qualitative properties of the system.In the recent years,splitting methods have attracted more and more interest.The purpose of this thesis is to develop splitting methods which can preserve the oscillatory properties of the solution of differential equations.Furthermore,the newly constructed splitting integrators will be applied to the numerical simulation of biological oscillators.The thesis is divided into four chapters.In Chapter 1,as the preliminary of the thesis,we present the background and elementary concepts of numerical methods for ordinary differential equations.The Lie-Trotter splitting of order one,Strang splitting of order two and Strang-splitting-based triple jump are introduced.The order conditions for general splitting methods are present.Two biological oscillators-the prey-predator system and the gene regulatory networks-are modeled in differential equations.Chapter 2 is devoted to the phase properties-dispersion and dissipation-of splitting methods.We prove that the Lie-Trotter splitting and the Strang splitting are dispersive of order two and the Strang-splitting-based triple jump is dispersive of order four.All the three splitting methods are zero-dissipative.We further derive the phase fitted Lie-Trotter split-ting,Strang splitting and triple jump.Numerical stability of the new methods is analyzed.The numerical results show that the new phase fitted methods are more efficient than their prototype splitting methods and Runge-Kutta methods of the same algebraic order.In chapter 3,exponential transformation and the time-freezing technique are utilized to construct the exponential Lie-Trotter splitting method and the exponential Strang splitting method.Their superiority is verified by numerical experiments.In chapter 4,based on the idea of blending the classical Runge-Kutta methods with splitting,the splitting Runge-Kutta methods are developed and investigated.The order conditions up to order three are presented.Two second-order and one third-order splitting Runge-Kutta methods are derived.Numerical results show that the new methods are much more efficient than the existed RK methods with the same order.Finally,we summarize the main contributions of this thesis and present the prospect for the future work.