二维分数阶非线性薛定谔方程的守恒数值方法

A conservative numerical method for fractional nonlinear Schrdinger equation in two dimensions

非线性薛定谔方程在许多领域有重要应用,尤其分数阶非线性薛定谔方程研究日益火热。主要研究二维分数阶非线性薛定谔方程的守恒数值求解方法。首先,为了减少存储量和运行时间,引入分数阶微分矩阵,应用加权和偏移Grunwald-Letnikov空间差分格式,对二维分数阶非线性薛定谔方程进行空间离散;然后,利用紧致隐式积分因子方法的优点(指数矩阵可以在预处理阶段计算和存储,在时间循环过程中可以直接应用,且对扩散项的精确计算与非线性项的隐式处理解耦,只需在每个时间周期内求解每个空间网格点的局部非线性代数方程组),对二维分数阶非线性薛定谔方程进行时间离散;最后,数值算例验证了方法的守恒性、准确性和有效性。

Nonlinear Schrdinger equation has important applications in many fields,especially the study on the fractional non-linear Schrdinger equation（FNLS）has been paying more and more attention.In this paper,we mainly discuss the conservation numerical solution of two-dimensional fractional nonlinear Schrdinger equation.Firstly,for two-dimensional fractional nonlinear Schrdinger equationon the basis of spatial discretization,the fractional differential matrix is introduced based on the weighted and shifted Grunwald-Letnikov（WSGD）space difference scheme to reduce the storage and CPU time.Secondly,for two-dimensional fractional nonlinear Schrdinger equation to develop the compact implicit integration factor（cIIF）method on time discretization,its advantage ofthe exponential matrix,which is a compact integral factor method,can be calculated and stored in the preprocessing stage and applied directly in the time loop.Another feature is the decoupling of the exact computation of diffusion terms from the implicit processing of nonlinear terms.The local nonlinear algebraic equations for each space grid point are solved in each time period.Finally,numerical examples are given to illustrate the conservation,accuracy and validity of the method.