Symmetric Exponential Integrators And Their Applications

The numerical solutions of differential equations have received much attention of many scholars and experts in the fields of hydrodynamics,optics,thermal physics and aerospace.Many classical algorithms have been studied and proposed such as symmetric methods and symplectic methods,Runge-Kutta methods,and so on.Many practical problems encountered in scientific research and engineering can be transformed into a system of differential equations and most of which are nonlinear differential equations.Therefore,it is of great significance to study the numerical solutions of nonlinear Hamiltonian systems.In this thesis,we study the symmetric exponential integrators and their applications.This thesis is divided into four chapters.Chapter 1 is the introduction of the thesis where we will introduce the present research status of symmetric exponential integrators for solving the nonlinear Hamiltonian systems and the charged-particle dynamics.In Chapter 2,we explore efficient symmetric and symplectic exponential integrators for solving the initial value problems.We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge-Kutta methods.Based on these conditions,some symmetric and symplectic exponential integrators up to order four are derived.Two numerical experiments are carried out and the results demonstrate the remarkable numerical behavior of the new exponential integrators in comparison with some symmetric and symplectic Runge-Kutta methods in the literature.In Chapter 3,we are devoted to the construction and analysis of symmetric exponential integrators for solving charged-particle dynamics.We derive and analyze symmetric exponential integrators for charged-particle dynamics in a strong and constant magnetic field.We first present the scheme of exponential integrators and then establish the symmetry conditions for the methods.Explicit symmetric exponential integrators up to order four are constructed on the basis of the symmetry conditions.In order to show the remarkable performances of new symmetric methods in comparison with symmetric Runge-Kutta methods and the 2-stage Gauss method,two numerical experiments are carried out and the numerical results demonstrate the super numerical behavior.Chapter 4 is about the summary of this article.