非线性空间分数阶Ginzburg-Landau方程的隐显型差分格式  

Implicit-explicit difference schemes for the nonlinear fractional Ginzburg-Landau equation

作　　者：王鹏德 黄乘明[2,3] Pengde Wang;Chengming Huang

机构地区：[1]河南财经政法大学数学与信息科学学院,郑州450046 [2]华中科技大学数学与统计学院,武汉430074 [3]华中科技大学工程建模与科学计算湖北省重点实验室,武汉430074

出　　处：《中国科学：数学》2020年第10期1505-1524,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11771163和11801140);中国博士后科学基金(批准号:2017M620591)资助项目。

摘　　要：本文对带Riesz分数阶导数的非线性空间分数阶Ginzburg-Landau方程引入一类二阶带权隐显型差分格式.该类格式在时间上对方程中的线性项采用隐式离散并对非线性项采用显式离散,同时在空间上采用四阶拟紧差分格式逼近Riesz分数阶导数.通过引入并调节权因子θ∈[1/2,1],可获得不同的隐显型格式,该类格式在每一时间步仅需求解一个系数矩阵与时间层无关的线性方程组.本文利用离散能量方法和G稳定性思想证明格式在lh2范数、Hhα/2半范数和lh∞范数意义下的无条件收敛性,且该证明对所有θ∈[1/2,1]一致成立.最后在数值测试中验证格式的数值精度,并比较当θ取不同值时所得格式在有限时间和长时间数值仿真中的有效性.This paper proposes and analyzes a family of second-order weighted implicit-explicit difference schemes for the nonlinear complex fractional Ginzburg-Landau equation involving the Riesz space fractional derivative.The schemes are implicit in the linear term in the underlying equation while explicit in the nonlinear term,and a fourth-order fractional quasi-compact method is used to approximate the Riesz fractional derivative.By introducing a factorθ∈[1/2,1]in the scheme,we can obtain different schemes by changing the factor.The schemes can be performed effectively because only a linear system with a coefficient matrix independent of the time level needs to be solved at each time step.We prove the unconditional convergence of the schemes in the lh2 norm,semi-Hhα/2 norm and l∞hnorm with a unified fashion for allθ∈[1/2,1]by using the discrete energy method and the idea of the G-stability.Numerical tests are provided to confirm the theoretical results.

关 键 词：空间分数阶Ginzburg-Landau方程 Riesz分数阶导数 隐显型方法 紧差分格式 逐点误差估计 

分 类 号：O241.3[理学—计算数学]