题 目 ：Efficient difference schemes for the fractional Ginzburg-Landau equation

报告人：河南财经政法大学数学与信息科学学院 王鹏德副教授

时 间：2017年9月22日上午

地 点：数学楼第一报告厅

摘 要：In this talk, we present an efficient difference scheme for the nonlinear complex Ginzburg-Landau equation involving the fractional Laplacian. The scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted Grunwald difference operator for the spatial fractional Laplacian.This scheme is second-order in both time and space. Our focus is on a rigorous theoretical analysis for the scheme.In order to overcome the difficulty caused bythe nonlocal property of the fractional Laplacian,we make a detailed study of the fractional approximation operator. The discrete fractional Gagliardo-Nirenberg inequality and an equivalence relation between an energy norm and the fractional Sobolev semi-norm are established. Then the scheme is proved to be unconditionally convergent in the l^2 norm with optimal order, in the sense that no restriction on the temporal step size in terms of the spatial discretization parameter needs to be assumed. Finally,numerical examples are given to validate the theoretical results and the effectiveness of the scheme.