High Accurate And Conservative Finite Difference Schemes For Nonlinear Schr(?)dinger Equation

Nonlinear Schr?dinger (NLS) equations play a very important role in the research of such aspects as high-energy physics, quantum mechanics, nonlinear optics, superconduction and deep ripples, etc. This thesis carries on research to the nonlinear Schr?dinger Equations. In this thesis, a linearizing conservative finite difference scheme is proposed for generic nonlinear Schr?dinger equations at first. The scheme conserves the energy and charge of systems, and its convergence and stability are proved. The scheme are showed to possess second order accuracy and fourth order accuracy in maximum norm for time and space respectively. Besides as the scheme is linearizing without iterative, its runtime is improved too. Then we study a class of nonlinear Schr?dinger equation involving quintic term. And a two-level scheme and a three-level scheme are proposed for the above problems. Both theory and numerical test results show the good stability, convergence and accuracy. At last, two high accurate and conservative finite difference Schemes are proposed for the radial symmetric nonlinear Schr?dinger equation in the same method. By means of numerical computation, it is followed that the new schemes are efficient.