Additive Runge-Kutta-Type Methods For Hamilton Systems

A large number of problems in physics,chemistry,biology and celestial mechanics can often be attributed t.o first-order or second-order differential equations.However,for many differential equations,the vector field is split into two or more parts,such as stiff and non-stiff,linear and non-linear.In order to effectively maintain the qualitative properties of the system,additive methods apply an appropriate numerical methods to each sub-field.This article aims to construct additive symplectic Runge-Kutta methods for solving first-order differential equations and additive symmetric Runge-Kutta-Nystr(?)m methods for solving second-order and to utilize them to solve Hamiltonian systems.This article is divided into four chapters.The first chapter provides preliminary knowledge,including the research background and the related concepts of the numerical solution of differential equations.Runge-Kutta(RK)methods for solving first-order differential equations,Runge-Kutta-Nystr(?)m(RKN)methods for second-order differential equations,and generalized additive Runge-Kutta methods for first-order split differential equations are introduced.Then rooted tree theory and order conditions for numerical methods are given.For Hamilton systems,we introduce the structure-preservation of RK methods and RKN methods,including the symplecticity and symmetry.The second chapter investigates general additive Runge-Kutta(GARK)methods for first-order Hamilton systems.The rooted trees and the related functions are discussed.On this basis,the first to fourth order conditions are presented.Then the symmetry of the method is studied.On this basis,(1,2)stage,2nd order,(1,4)stage,2nd order,(2,3)stage 4th order and(3.4)stage 4th order symmetric GARK(SGARK)methods are constructed,and their numerical stability is then analyzed.Numerical experiments show that the new SGARK methods are more efficient than the nonsymmetric GARK methods in the literature.Chapter three adds a symplecticity condition to symmetric GARK methods and constructs symmetric and symplectic GARK(SSGARK)methods.The stability of the new SSGARK methods is analyzed and their stability regions are depicted.The results numerical experiments show that compared with non-symplectic GARK methods,the new SSGARK methods can preserve the Hamiltonian more precisely.The fourth chapter studies generalized additive Runge-Kutta-Nystr(?)m(ARKN)methods for the second-order split systems.Order conditions are derived via bicolored rooted tree theory.The symmetry condition is proved.With the order conditions and the symmetry condition,(1,2)stage and 2nd order,(2,2)stage and 4th order,(2,3)stage and 2nd order symmetric GARKN(SGARKN)methods are constructed.Also the stability of SGARKN methods is analyzed.Numerical experiments show that the new SGARKN methods have higher accuracy than the traditional RKN methods.In short,this paper constructs the symmetric additive Runge-Kutta method for firstorder differential equations based on the order conditions and the symmetry condition.Secondly,a symplecticity condition is added to symmetric GARK methods,and symmetric and symplectic Runge-Kutta methods are constructed.Finally,the additive symmetric RungeKutta-Nystr(?)m methods for second-order differential equations are studied.Numerical simulation results show that after properly splitting of the vector field,the additive RK(N)methods have higher accuracy than traditional methods of the same order.