Zhu Xiaogang, Wan Haiyang, Zhang Yaping
School of Science, Shaoyang University, Shaoyang, 422000, Hunan, People's Republic of China.
School of Future Technology, South China University of Technology, Guangzhou, 510641, Guangdong, People's Republic of China.
Sci Rep. 2024 Oct 16;14(1):24257. doi: 10.1038/s41598-024-75547-2.
In this article, we propose a split-step finite element method (FEM) for the two-dimensional nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives in space. The space-fractional NLS is first spatially discretized by finite element scheme and the semi-discrete variational scheme is obtained. We prove that it maintains the mass and energy conservation laws. Then, we establish a fully discrete split-step finite element scheme for the considered problem, which avoids the iteration at each time layer, thereby significantly reducing computational cost. The discrete mass conservation property and error estimate of this split-step finite element scheme is derived. Finally, illustrative tests and the numerical simulation of dynamic of wave solutions are included to confirm its effectiveness and capability.
在本文中，我们针对具有空间Riesz分数阶导数的二维非线性薛定谔方程（NLS）提出了一种分步有限元方法（FEM）。首先通过有限元格式对空间分数阶NLS进行空间离散，得到半离散变分格式。我们证明它保持了质量和能量守恒定律。然后，针对所考虑的问题建立了全离散分步有限元格式，该格式避免了在每个时间层进行迭代，从而显著降低了计算成本。推导了该分步有限元格式的离散质量守恒性质和误差估计。最后，给出了说明性测试以及波动解动力学的数值模拟，以证实其有效性和性能。
