The Research Of Energy-Preserving Algorithms For Hamiltonian Systems

ABSTRACT:The practical application of differential equations is very extensive, they have been widely used in the fields of Celestial Mechanics, Chemistry and Biology, etc. Since only a few of differential equations can be obtained with exact solutions, it has great significance to study its numerical methods.About the numerical methods for differential equations, we have created some effective algorithms such as Euler algorithms, Adams algorithms, Runge-Kutta algorithms. Particularly due to the development of computer, more efficient algorithms have been discovered and used continuously. However, there hasn’t good stability and long-term tracking capacity when they are used to calculate some models. A basic idea of constructing numerical methods for differential equations is that they can preserve the important properties of differential equations as much as possible. The structure-preserving algorithms, created by this idea, have unique advantages in terms of stability and long-term numerical tracking capability.This article introduces the properties of Hamiltonian systems and symplectic geometric algorithms, we focus on the research of energy-preserving algorithms. The discrete gradient plays an important role in the construction of energy-preserving algorithms. For function fmg’, we study the construction of discrete gradients and give the constructing method. Meanwhile, the construction of discrete gradient for function H=gTAg are considered, and some relevant properties are given. For a specific example, some different discrete gradients are constructed and energy-preserving methods based on those discrete gradients are obtained. The numerical experiments show that the energy deviation of LM4method and RK4method is growing with the growth of time compared with the energy-preserving methods.