An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation

详细信息    查看全文

• 作者：Pengde Wang ^{pengde_wang@yeah.net" class="auth_mail" title="E-mail the corresponding author} ; Chengming Huang ; ^{chengming_huang@hotmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional Ginzburg&ndash ; Landau equation ; Fractional Laplacian ; Riesz fractional derivative ; Weighted and shifted Grü ; nwald difference ; Convergence
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：312
• 期：Complete
• 页码：31-49
• 全文大小：1990 K

文摘

This paper proposes and analyzes an efficient difference scheme for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted Grünwald difference operator for the spatial fractional Laplacian. By virtue of a careful analysis of the difference operator, some useful inequalities with respect to suitable fractional Sobolev norms are established. Then the numerical solution is shown to be bounded, and convergent in the $l_h^2$ norm with the optimal order O(τ²+h²)$O({\tau }^2+h^2)$ with time step τ and mesh size h. The a priori bound as well as the convergence order holds unconditionally, in the sense that no restriction on the time step τ in terms of the mesh size h needs to be assumed. Numerical tests are performed to validate the theoretical results and effectiveness of the scheme.