The Theoretical Analysis And Applications Of High Order Energy-preserving Average Vector Field Method

In1984, Feng kang academician and his study team proposed the symplectic geometric algorithms of the Hamilton system. Based on the symplectic geometric algorithms,Bridge and Reich et al have de-veloped multi-symplectic geometric algorithms of Hamilton system. Symplectic and multi-sumplectic geometric algorithms have more advantage in stability and long time computing than traditional nu-merical algorithms However,symplectic and multi-symplectic geometric algorithms nearly preserve the energy for the nonlinear Hamilton system. Recently, the second order average vector field (AVF) method preserving the Hamiltonian energy exactly is proposed by Quispel and McLachlan. Moreover,the second order AVF method has been widely applied to solve energy conservation partial differential equations.we mainly investigate the construction and analysis of high order energy-preserving methods by average vector field method and several types of important energy conservation partial differential equa-tions are solved by the high order energy-preserving methods.In chapter l,the second order average vector field method is applied to solve the coupled nonlinear Schrodinger equations. The comparison of the energy preserving between second average vector field method and symplectic geometric algorithms is made.In chapter2, In section1, based on the fourth order AVF method in time direction and finite dif-ference method in spatial direction, the high order energy preserving scheme of the strongly coupled nonlinear Schrodinger system is developed. Then,the proposed scheme is applied to simulate solitons evolution of the strongly coupled Schrodinger system. In section2, the semi-Hamilton system of the KdV equation is obtained by the pseudo-spectral method in spatial variable. Then the semi-Hamilton pseudo-spectral system is discretizating by the fourth order average vector flied method in time vari-able.Thus, a high order energy-preserving scheme of the KdV equation is derived.The evolution of the solitary wave are simulated by the new scheme. In section3, The energy-preserving property of the high order average vector field method is proved. A new third and fourth order average vector field method is derived by the Bootstrapping technique. A high order energy conservation scheme of the "good" Boussinesq equation is obtained by the fourth order average vector field derived by Quispel and McLachlan. In section4, based on the average vector field method, two third order and one fourth order energy-preserving scheme of the one dimensional sine-Gordon equation is developed. The developed schemes are used to simulate the solitary wave evaluation and test accuracy. In section5,the average vector field method are developed for solving the two dimensional partial differential equation. The high order energy-preserving scheme of the two dimensional Gross-Pitaevskii equation is obtained by the pseudo-spectral method and the fourth order average vector flied method. The evolution of the solitons is simulated by developed scheme. In section6, the two dimensional sine-Gordon is discretizated by the pseudo-spectral method in spatial variable and the fourth order average vector flied method in time variable.Thus, a high order energy-preserving scheme of the two dimensional sine-Gordon equation is obtained. The evolution of the elliptical ring soliton and elliptical breather soliton is simulated by the high order scheme.In chapter3, the fifth order saturated nonlinear Schrodinger equation is transformed into the multi-symplectic structure and discretizated by the middle Preissman scheme. The multi-symplectic scheme and the corresponding multi-symplectic conservation is obtained. The solitary waves with the different nonlinear saturated effects and the different amplitude are simulated by the multi-symplectic scheme.In chapter4, slow light and slow optical solitons have become the focus of quantum optics and non-linear optics for their application in all optical communication technology etc. Applying the fourth order compact splitting step finite difference method to discretizate the generalized nonlinear Schrodinger equation precisely describing the behaviors of high order dual steady slow optical solitons in three-level gaseous media, we obtain the corresponding discrete scheme. Taking the Rb atomic fine structure pa-rameters to simulate, we analyze the evolution behaviors of single and multiple slow optical solitons by changing the fine structure parameter and the initial probe field properly.