High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension
High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension
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DOI：
10.1007/s10915-018-0716-8
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发表时间：
2018-04
影响因子：
2.5
通讯作者：
V. Bokil;Yingda Cheng;Yan Jiang;Fengyan Li;Puttha Sakkaplangkul
中科院分区：
文献类型：
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作者：
V. Bokil;Yingda Cheng;Yan Jiang;Fengyan Li;Puttha Sakkaplangkul
在本文中，我们考虑电磁波在一维非线性光学介质中的传播。我们通过与时间相关的麦克斯韦方程组与非线性常微分方程（ODE）系统耦合来模拟电磁波的传播，用于介质对电磁波的响应。常微分方程中的非线性描述了瞬时的电子克尔响应和剩余的拉曼分子振动响应。常微分方程还包括单共振线性洛仑兹色散。对于这样的模型，我们将设计和分析具有任意（偶数）空间阶和二阶时间的全离散时域有限差分（FDTD）方法。完全离散方法的可证明稳定性是一个挑战，这取决于非线性项时间离散化的选择。在Boeker等人（J Comput Phys 350：420-452，2017）中，我们在不连续Galerkin方法的背景下提出了二阶蛙跳和梯形时间格式的新修改，以离散该麦克斯韦模型中的非线性项。在这里，我们继续这项工作，开发类似的时间离散的FDTD方法的框架内。更具体地说，我们设计了完全离散的修改蛙跳FDTD方法，被证明是稳定的适当CFL条件下。这些方法可以看作是Yee-FDTD方法在该非线性麦克斯韦模型中的推广。我们还设计了完全离散的梯形FDTD方法，被证明是无条件稳定的。通过数值实验，包括扭结，反扭结波和三次谐波产生孤子传输的全离散FDTD方法的性能进行了证明。
In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time-dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leap-frog FDTD methods which are proved to be stable under appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD methods which are proved to be unconditionally stable. The performance of the fully discrete FDTD methods are demonstrated through numerical experiments involving kink, antikink waves and third harmonic generation in soliton propagation.