| Schrodinger equation is one of the most important models of mathematical physics with applications to different fields such as plasma physics, nonlinear op-tics, water waves, biomolecule dynamics and many other fields. Many scientists are devoted to studying the exact and numerical solution of the Schrodinger equa-tion. This thesis mainly considers the Crank-Nicolson method and high-order compact difference method based on the Richardson's extrapolation technique for one and two-dimensional linear Schrodinger equations.This work consists four sections. Section 1 is preface. we introduce research background, purpose and significance, and describe the research situation of nu-merical solution for Schrodinger equation. Finally, the organizational structure of this work is given.In section 2, we present the Crank-Nicolson scheme for solving one and two-dimensional linear Schrodinger equations. Then Richardson's extrapolation method is successfully applied to the scheme and the approximate solution with high accuracy is gained. This method is shown to be unconditionally stable. In this paper, we proof the stability, and finally make a numerical experiment and compared with the work before done.In section 3, a high-order compact difference method based on the Richard-son extrapolation technique is proposed to solve one and two-dimensional lin-ear Schrodinger equations. For a particular implementation, firstly, numerical results are obtained on the fourth-order compact difference formulas for the sec-ond derivatives. Then, the Richardson extrapolation method is used to get an accuracy solution for the Schrodinger equation, which is fourth order in space and fourth order in time. It is proved to be unconditionally stable by Fourier analysis. Numerical results obtained have compared with the exact one. It is shown that numerical experiments are made to demonstrate the high accuracy and validity of this method.Section 4 is conclusion, we make a conclusion on the whole work.
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