A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation
A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation
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DOI：
10.3934/nhm.2023048
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发表时间：
2023
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影响因子：
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通讯作者：
Tingting Ma;Yayun Fu;Yuehua He;Wenjie Yang
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文献类型：
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作者：
Tingting Ma;Yayun Fu;Yuehua He;Wenjie Yang
本文提出了一种求解分数阶非线性薛定谔方程的新方法。我们的方法结合了不变能量二次化方法和指数时间差分方法，得到了一个线性隐式的能量保持方案。为了实现这一点，我们引入了一个辅助变量，推导出一个等效系统与修改后的能量守恒定律。该格式采用稳定指数时间差分近似进行时间积分，空间采用傅立叶伪谱离散，得到线性隐式全离散格式。与原有的能量保持指数积分方案相比，我们的方法是更有效的，因为它不需要非线性迭代。数值实验证实了该方案在能量守恒方面的有效性和在长时间计算中的效率。
In this paper, we present a new method to solve the fractional nonlinear Schrödinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.