分数阶非线性Schr?dinger方程的时间分裂算法

Time-splitting Method for the Fractional Nonlinear Schr?dinger Equation

主要研究分数阶非线性Schr?dinger方程的时间分裂算法,将分数阶非线性Schr?dinger方程分裂成一个线性方程和一个非线性方程分别求解。其中,非线性方程可精确求解,并满足"点点守恒",而线性方程利用Crank-Nicolson差分格式离散求解。证明了该算法在离散形式下保持了原方程的质量和能量的守恒性,是无条件稳定的,收敛误差为O(h^2+τ~2)。最后通过数值实验验证了该算法的可行性和精度,说明该算法是一种简单有效的算法。

A time-splitting method for solving the fractional nonlinear Schrodinger equation with space fractional derivative was proposed.In this method,the fractional nonlinear Schrodinger equation was split into a linear equation and a nonlinear equation,where the nonlinear equation could be solved exactly and satisfy4conservation of point7,and the linear equation could be solved by using Crank-Nicolson discretization method.The conservation of mass and energy for the original equation was kept in this method.The unconditional stability and the convergence with the truncation error0(h2+r2)were proved.Finally,numerical examples were presented to show that the method was both effective and accurate,which indicated that the method was simple and effective.