A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation
A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation
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DOI：
10.1137/140961560
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发表时间：
2014-12
期刊：
影响因子：
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通讯作者：
Xuan Zhao;Zhi‐zhong Sun;Zhao-peng Hao
中科院分区：
文献类型：
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作者：
Xuan Zhao;Zhi‐zhong Sun;Zhao-peng Hao
本文给出了一种新的逼近Riesz导数的紧算子，证明了该紧算子具有四阶精度。结合空间离散中的紧算子，提出了求解二维非线性空间分数阶薛定谔方程的线性化差分格式。证明了该差分格式是唯一可解、稳定和收敛的，阶为$O(tau^2+h^4)$，其中$tau$是时间步长，$h=max h1，h2$，以及$h1，h2$分别是x$方向和y$方向的空间网格尺寸。在线性化差分格式的基础上，提出并分析了一种紧致交替方向隐式格式。数值结果表明，紧致算子没有带来额外的计算代价，但大大提高了格式的精度。
In this paper, a novel compact operator is derived for the approximation of the Riesz derivative with order $\alpha\in(1,2].$ The compact operator is proved with fourth-order accuracy. Combining the compact operator in space discretization, a linearized difference scheme is proposed for a two-dimensional nonlinear space fractional Schrodinger equation. It is proved that the difference scheme is uniquely solvable, stable, and convergent with order $O(\tau^2+h^4)$, where $\tau$ is the time step size, $h=\max\{h_1,h_2\}$, and $h_1,\,h_2$ are space grid sizes in the $x$ direction and the $y$ direction, respectively. Based on the linearized difference scheme, a compact alternating direction implicit scheme is presented and analyzed. Numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly.