Research And Application Of Runge - Kutta Method For High - Order Diagonal Implicit

Hamiltonian systems often arise in different fields of applied sciences such as physical mechanics, celestial mechanics, astrophysics, chemistry, electronics, molecular dynamics,and so on. In the past decades, many research was done to study the property of the Hamiltonian systems, and numerical methods were constructed on basis of these special property.Symplecticness which was introduced by Kang Feng is a very important property of the systems, and lots of works were done about it.In this thesis, we do some research on the construction of high order diagonally implicit symplectic Runge-Kutta method and its application. Based on the symplectic and symmetrical properties, we construct a series of fifth-order diagonally implicit symplectic RungeKutta method with six stages and a sixth-order diagonally implicit symmetric and symplectic Runge-Kutta method with seven stages. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing Runge-Kutta methods in scientific literature. The attainable algebraic order of the diagonally implicit symplectic Runge-Kutta method and the diagonally implicit symmetric symplectic Runge-Kutta method are investigated.We investigate the A-stability and P-stability of the proposed method. The proposed methods were proved to be A-stable while p ≤ 2, and not A-stable while p ≥ 3. The proposed methods were proved to be P-stable.In the last of the paper, we investigate the numerical integration of Hamiltonian systems with oscillating solutions. A diagonally implicit symplectic Runge-Kutta method with high algebraic order and high dispersion orderis presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.