Study Of Structure Preserving Algorithm Of "Good"Boussinesq Equation And Diffusion Equation

Structure preserving algorithm, which is a important branch of computational mathematics, is formed based on the symplectic geometry algorithm proposed by the famous computational mathemati-cian Fengkang academician and his study group. The main methods of structure preserving algorithm have the symplectic geometry algorithm, the multi-symplectic method, Lie group method and the en-ergy preserving method, et al. Structure preserving algorithm has been widely applied to astronomy, quantum physics, plasma physics, nonlinear optics, atmosphere, fluid mechanic,et al. Recently, energy preserving method has gain great success. Different energy preserving methods have been proposed. R.I.McLachlan, G.R.W.Quispel et al proposed the average vector field method based on the discrete gra-dient method, which is a class of energy preserving method. At present, few people study the theory and application of the high order average vector field method.This thesis mainly divided into three parts:In chapter1, we ananlyze the accuracy of the average vector field method by the Taylor series and B series theory. The energy conservation property of the Hamiltonian system,which can be preserved by the high order average vector field method, has also been proved.In chapter2, a new average vector field method is proposed to solve the "good" Boussinesq equa-tion. The semi-discrete system of the "good" Boussinesq equation obtained by the pesudo-spectral method in spatial variable is discretizated by the average vector field method. Thus, the average vec-tor field scheme of the "good" Boussinesq equation is proposed. The"good" Boussinesq equation is simulated by the average vector field scheme with different amplitudes. Numerical results show that the average field scheme can have excellent performance in simulating the solitary waves evolution behaviors and persevere the energy of the equation exactly.In chapter3, Lie group method is a class of important method in structure preserving algorithm. The diffusion equation is discretized in spacial direction and transformed into the ordinary differential equations. The ordinary differential equations are solved by the Lie group method and the explicit Runge-Kutta method. Numerical results show the Lie group method is more stable than the corresponding explicit Runge-Kutta method.