Two Finite Difference Schemes For A Schrodinger Equation With Wave Operator

Nonlinear Schrodinger(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 a class of nonlinear Schrodinger Equation with wave operator of initial value periodic problem. In this thesis, a semi-discrete finite difference scheme is presented at first, then a discrete-time difference orthogonal spline collocation scheme is proposed for the above problem by using piecewise pth Hermite interpolation combined with finite difference method. This thesis proves two schemes'convergence and stability. In the course of solving semi-discrete scheme, it uses four-order Runge-Kutta method and adapted Runge-Kutta method. A large number of numerical experiments show that, besides having high accuracy , being easy to realize and stable, the adapted Runge-Kutta method can obtain error directly, auto-adjustment for step size without loss of precision. The difference orthogonal spline collocation scheme are showed to possess second order accuracy in maximum norm and fourth order in L~2 -norm for time and space respectively. Implementations of this scheme are discussed and numerical results are also presented which demonstrate the results in theory.