非线性阻尼薛定谔方程的高阶紧致有限差分算法
A High-Order Compact Finite Difference Scheme for the Nonlinear Damped Schrödinger Equation
DOI: 10.12677/aam.2026.153098, PDF,    国家自然科学基金支持
作者: 薛志超, 云丹瑶：内蒙古师范大学数学科学学院，内蒙古 呼和浩特；何斯日古楞：呼和浩特民族学院数学与大数据学院，内蒙古 呼和浩特；王桂霞*：内蒙古师范大学数学科学学院，内蒙古 呼和浩特；内蒙古自治区应用数学中心，内蒙古 呼和浩特；无穷维哈密顿系统及其算法应用教育部重点实验室，内蒙古 呼和浩特
关键词: 薛定谔方程；阻尼；紧致差分格式；收敛性；误差估计；Nonlinear Schrödinger Equation； Damped； Compact Finite Difference Scheme； Convergence； Error Estimates
摘要: 非线性阻尼薛定谔方程刻画光脉冲在吸收型非线性光纤中的传输等有着广泛应用。本文基于阻尼方程空间六阶紧致差分离散，结合阻尼项的分裂方程和Strang分裂法，构造了一种高阶紧致差分格式。利用能量估计法分析了所提格式的能量和质量的衰减性质。进而，证明了所提格式的时间二阶、空间六阶误差估计结果。最后，通过数值算例验证了该格式的有效性与可靠性。
Abstract: The nonlinear damped Schrödinger equation has been widely applied in modeling the propagation of optical pulses in absorptive nonlinear optical fibers. In this paper, a high-order compact finite difference scheme is constructed by employing a sixth-order compact finite difference discretization in space for the damped equation, combined with a splitting formulation of the damping term and the Strang splitting method. The energy and mass decay properties of the proposed scheme are analyzed using the energy estimate method. Furthermore, second-order accuracy in time and sixth-order accuracy in space are rigorously established. Finally, numerical experiments are presented to verify the effectiveness and reliability of the proposed scheme.
文章引用：薛志超, 云丹瑶, 何斯日古楞, 王桂霞. 非线性阻尼薛定谔方程的高阶紧致有限差分算法[J]. 应用数学进展, 2026, 15(3): 188-200. https://doi.org/10.12677/aam.2026.153098

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