Geodesic
A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations
Applications of Mathematics, Tome 66 (2021) no. 2, pp. 213-232

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We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma $ in time, where $0\gamma 1$ is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.
DOI : 10.21136/AM.2020.0339-19
Classification : 65M06, 65M20, 65M60
Keywords: nonlinear time fractional Schrödinger equations; $L1$ formula; hybrid compact difference method; linearization; unconditional stability 
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     title = {A spatially sixth-order hybrid $L1${-CCD} method for solving time fractional {Schr\"odinger} equations},
     journal = {Applications of Mathematics},
     pages = {213--232},
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Zhang, Chun-Hua; Jin, Jun-Wei; Sun, Hai-Wei; Sheng, Qin. A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations. Applications of Mathematics, Tome 66 (2021) no. 2, pp. 213-232. doi: 10.21136/AM.2020.0339-19

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