Efficient Numerical Schemes For Nonlinear Schr(?)dinger Equation In Two Dimensions

Nonlinear Schr?dinger equations are widely used in many fields such as physics and engineering.In particular,the study of nonlinear fractional Schr?dinger equations has attracted more and more scholars’ attention.In this paper,we first study the two-dimensional nonlinear integer order Schr?dinger equation,apply the second-order finite difference method to the spatial discretization,and perform orthogonal decomposition on the discrete difference matrix.The discrete Fourier transform can be used to realize the product of the matrix.In time discretization,the second-order compact implicit integration factor method is applied,and the fast Fourier transform is combined to increase the computation efficiency.The numerical solution is obtained by Picard iteration on each grid.Compact implicit integration factor method,compared with the non-compact implicit integration factor method can reduce the storage and improve the computation speed.In this paper,two-dimensional nonlinear fractional Schr?dinger equations are also studied.The second-order weighted shift Grünwald–Letnikov difference(WSGD)method is used to discretize the space.The obtained differential matrix is a real-valued symmetric positive definite matrix with Cholesky decomposition.It is very useful to prove the discrete conservation law,which proves the mass and energy conservation of semi-discrete forms.Two methods are proposed on time discretization.One is based on the Crank-Nicolson method,which can prove that it can maintain the full discrete mass and energy conservation.The other is the compact implicit integral factor method with good stability.Finally,numerical results are given to prove the conservation,accuracy and effectiveness of the proposed method.The thesis is divided into five parts:The first chapter introduces the research background of the equation,the research status,the innovation point and the main work of this paper;the second chapter introduces the definition of fractional derivative,the numerical method and the WSGD operator;the third chapter studies the conservation numerical method of two dimensional nonlinear Schr?dinger equation.The numerical method is based on finite difference method for spatialdiscretization.The compact implicit integral factor method is applied to time discretization.Combined with Fast Fourier transform,the numerical results verify the effectiveness of the method.In the fourth chapter,the effective difference schemes of two-dimensional nonlinear fractional Schr?dinger equations are proposed and analyzed.Based on the WSGD difference operator,two time discretization methods are proposed,one based on the Crank-Nicolson method and the other is the compact implicit integration factor.Numerical results are given to prove the conservation,convergence and effectiveness of the proposed method.Finally,the above contents are summarized and the next research direction is pointed out.