保持结构的高阶FDTD方法用于具有洛伦兹色散的非线性Kerr-Debye模型
Structure Preserving High-Order FDTD Methods for the Nonlinear Kerr-Debye Model with Lorentz Dispersion
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中文总结 AI 辅助
本文针对非线性Kerr-Debye模型提出保持结构的高阶FDTD方法，结合交错空间离散与蛙跳/梯形时间积分，实现能量稳定与正性保持，并高效模拟孤子传播等非线性现象。
中文摘要 AI 辅助
我们针对非线性光学介质中超短脉冲传播的非线性色散电磁模型，在时域有限差分（FDTD）框架下开发并分析了若干族全离散高阶方法。该模型由时域麦克斯韦方程组耦合两个常微分方程组成：一维空间中的非线性Kerr-Debye方程和线性色散洛伦兹方程。所提出的方法将$2M$阶（整数$M>0$）的交错空间离散化与二阶显式蛙跳或隐式梯形时间积分器相结合。为了处理非线性Kerr-Debye方程中的刚性松弛，我们采用了文献[PengJCP2020]中引入的修正指数积分器，该积分器在松弛时间趋于零的极限下确保渐近保持（AP）行为。我们证明了所得格式满足连续能量恒等式的离散类似物，并保持非线性磁化率的正性（PP）。对于基于蛙跳时间积分器的格式，我们在CFL条件下建立了能量稳定性，而基于梯形时间积分器的格式是无条件能量稳定的。基于梯形时间积分器的格式通过Richardson外推进一步实现了时间上的高阶精度，在空间和时间上均达到$2M$阶精度。数值实验证明了理论收敛速度以及所提出格式捕捉非线性波动现象（如孤子传播、自陡峭和奇次谐波产生）的能力。这项工作将先前在间断伽辽金框架中的AP-PP和能量稳定方法扩展到FDTD设置，为模拟非线性色散光学介质中的电磁波传播提供了一种计算高效、灵活且稳健的工具。
英文摘要
We develop and analyze several families of fully discrete, high-order methods in the finite difference time domain (FDTD) framework for a nonlinear dispersive electromagnetic model for ultrashort pulse propagation in a nonlinear optical medium. The model consists of the time domain Maxwell's equations coupled to two ordinary differential equations; the nonlinear Kerr-Debye equation and the linear dispersive Lorentz equation, in one space dimension. The proposed methods combine staggered spatial discretizations of $2M$th-order, for integer $M>0$, with second-order explicit leapfrog or implicit trapezoidal time integrators. To handle the stiff relaxation in the nonlinear Kerr-Debye equation, we adopt a modified exponential integrator, introduced in \cite{PengJCP2020}, which ensures asymptotic-preserving (AP) behavior in the limit of vanishing relaxation time. We prove that the resulting schemes satisfy a discrete analogue of the continuous energy identity and preserve the positivity (PP) of the nonlinear susceptibility. For the schemes based on the leapfrog time integrator, we establish energy stability under a CFL condition, while the schemes based on the trapezoidal time integrator are unconditionally energy stable. High-order temporal accuracy in the trapezoidal time integrator based schemes is further achieved via Richardson extrapolation yielding $2M$th-order accurate methods in both space and time. Numerical experiments demonstrate the theoretical convergence rates and the ability of the proposed schemes to capture nonlinear wave phenomena such as soliton propagation, self-steepening, and odd harmonic generation. This work extends previous AP-PP and energy stable approaches in a discontinuous Galerkin framework to the FDTD setting, offering a computationally efficient, flexible and robust tool for simulating electromagnetic wave propagation in nonlinear dispersive optical media.
发表机构
- Oregon State University（俄勒冈州立大学）
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