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A second-order implicit difference scheme for the nonlinear time-space fractional Schrödinger equation
Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2020-07-01 , DOI: 10.1016/j.apnum.2020.03.004 Mingfa Fei , Nan Wang , Chengming Huang , Xiaohua Ma
Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2020-07-01 , DOI: 10.1016/j.apnum.2020.03.004 Mingfa Fei , Nan Wang , Chengming Huang , Xiaohua Ma
Abstract In this paper, we develop an implicit difference method for solving the nonlinear time-space fractional Schrodinger equation. The scheme is constructed by using the L2- 1 σ formula to approximate the Caputo fractional derivative, while the weighted and shifted Grunwald formula is adopted for the spatial discretization. The stability and unique solvability of the difference scheme are analyzed in detail. Moreover, we prove that the numerical solution is convergent with second-order accuracy in both temporal and spatial directions. Finally, a linearized iterative algorithm is provided and some numerical tests are presented to validate our theoretical results.
中文翻译:
非线性时空分数薛定谔方程的二阶隐式差分格式
摘要 在本文中,我们开发了一种求解非线性时空分数阶薛定谔方程的隐式差分方法。该方案采用L2-1σ公式逼近Caputo分数阶导数,而空间离散采用加权移位Grunwald公式。详细分析了差分格式的稳定性和唯一可解性。此外,我们证明了数值解在时间和空间方向上都以二阶精度收敛。最后,提供了一种线性化迭代算法,并提出了一些数值测试来验证我们的理论结果。
更新日期:2020-07-01
中文翻译:
非线性时空分数薛定谔方程的二阶隐式差分格式
摘要 在本文中,我们开发了一种求解非线性时空分数阶薛定谔方程的隐式差分方法。该方案采用L2-1σ公式逼近Caputo分数阶导数,而空间离散采用加权移位Grunwald公式。详细分析了差分格式的稳定性和唯一可解性。此外,我们证明了数值解在时间和空间方向上都以二阶精度收敛。最后,提供了一种线性化迭代算法,并提出了一些数值测试来验证我们的理论结果。