Preconditioned Iterative Methods Based On PSS For Fractional Nonlinear Schr(?)dinger Equations

Recently,problems with fractional derivatives are appeared widely in physics applications and mathematics theory,among which the fractional nonlinear Schr(?)dinger equation is one of the most important equations.In this paper,we consider the numerical solutions for the fractional nonlinear Schr(?)dinger equation.By using the linearly implicit conservative difference scheme,we obtained a series of complex linear equations which need to be solved numerically.The main contribution of this paper is how to design efficient preconditioned iterative methods for such complex linear systems.The details are as follows:(1)Firstly,we consider the uncoupled fractional nonlinear Schr(?)dinger equation.Using the linearly implicit conservative difference scheme leads to a series of complex linear system.The coefficient matrix can be written as the sum of a non-negative diagonal matrix,a positive symmetric Toeplitz matrix and a complex unit matrix.Based on the structure of the coefficient matrix,we split it into positive definite and semi-positive definite parts.Then we propose a class of iterative methods based on alternating direction technique,and prove the convergence of the new iterative method theoretically.(2)Based on the positive definite and semi-positive definite splitting,we propose the corresponding preconditioner.In practical applications,in order the reduce the computational cost,we use the circulant matrix to approximate the involved Toeplitz matrix.(3)Finally,we consider the coupled fractional nonlinear Schr(?)dinger equation.By making use of the structure of the coefficient matrix,we propose a class of iterative methods based on positive definite and semi-positive definite splitting,as well as the corresponding preconditioner.Numerical results show that the new precoditioner has very good performance.