The General Block Method For Linear Hamiltonian Systems

Hamiltonian systems have important applications in many fields,such as mechanics,optics,plasma physics,pharmacology and so on.It has many properties and the most important one is the conservation of the area and volume in the phase space.When numerically solving the Hamiltonian systems,it is hoped that the numerical solutions obtained by the numerical methods will preserve the geometric construction of the exact solutions.In this thesis,we are concerned with the applicability of block methods for the approximation of the solution of Hamiltonian systems,and obtain sufficient conditions for the block methods to preserve symplecticity and quadratic form.First,we recall the background of the Hamiltonian systems,and introduce the basic theory of Runge-Kutta methods and the block methods,and briefly summarize the research results in this field.Second,we propose general block methods for solving the Hamiltonian systems and provide sufficient conditions under which these methods can preserve symplecticity and quadratic form.In particular,we will study the applicability of the block ?-methods.At last,some symplectic general block methods and block ?-methods are given.Three points and four points block methods are used to solve linear and nonlinear Hamiltonian systems.Numerical results confirm our theoretical results for linear problems and show that these methods are also effective for solving nonlinear Hamiltonian systems.