A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrodinger equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:358

Article

Li, Meng¹  Gu, Xian-Ming^2,3  Huang, Chengming^4,5  Fei, Mingfa⁴  Zhang, Guoyu⁴ 

[1] Zhengzhou Univ, Sch Math & Stat, Zhengzhou 450001, Henan, Peoples R China

[2] Southwestern Univ Finance & Econ, Sch Econ Math, Chengdu 611130, Sichuan, Peoples R China

[3] Univ Groningen, Johann Bernoulli Inst Math & Comp Sci, Nijenborgh 9,POB 407, NL-9700 AK Groningen, Netherlands

[4] Huazhong Univ Sci & Technol, Sch Math & Stat, Wuhan 430074, Hubei, Peoples R China

[5] Huazhong Univ Sci & Technol, Hubei Key Lab Engn Modeling & Sci Comp, Wuhan 430074, Hubei, Peoples R China

关键词: Coupled nonlinear fractional Schrodinger equations;    Finite element method;    Mass and energy conservation;    Unconditional convergence;    Circulant preconditioner;    Krylov subspace methods;   

DOI  :  10.1016/j.jcp.2017.12.044

来源: Elsevier

【 摘 要 】

In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schrodinger equations. We prove that the scheme preserves both the mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical induction, the discrete scheme is proved to be unconditionally convergent in the sense of L-2-norm and H-alpha/2-norm, which means that there are no any constraints on the grid ratios. Then, the prior bound of the discrete solution in L-2-norm and L-infinity-norm are also obtained. Moreover, we propose an iterative algorithm, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners. This method can reduce the memory requirement of the proposed linearized finite element scheme from O(M-2) to O(M) and the computational complexity from O(M-3) to O(M log M) in each iterative step, where M is the number of grid nodes. Finally, numerical results are carried out to verify the correction of the theoretical analysis, simulate the collision of two solitary waves, and show the utility of the fast numerical solution techniques. (c) 2018 Elsevier Inc. All rights reserved.

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