一维非线性材料间断Helmholtz方程的高阶数值方法
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension
摘要：
非线性亥姆霍兹方程（NLH）模拟了电磁波在克尔介质中的传播，描述了非线性光学等领域的一系列重要现象。在我们以前的工作中，我们开发了一个四阶方法的数值解，涉及一个迭代求解器的基础上冻结的非线性。该方法使非傍轴制度的非线性自聚焦的直接模拟，和后向散射的定量预测。然而，我们的模拟表明，非线性的大小存在一个阈值，超过该阈值，迭代就会出现分歧。在这项研究中，我们数值求解一维NLH使用牛顿型非线性求解器。由于克尔非线性包含场的绝对值，因此NLH必须重新转换为两个实方程的系统，以便应用牛顿方法。我们的数值模拟表明，牛顿的方法收敛迅速，并在对比与基于冻结的非线性迭代，使计算非常高的非线性。此外，我们还对非线性线性方程组引入了一种新的紧致有限体积四阶离散方法，一维结果为以后分析多维问题奠定了基础。
英文摘要：
The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge. In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton's method. Our numerical simulations show that Newton's method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity. In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities.The one-dimensional results of the current paper create a foundation for the analysis of multi-dimensional problems in the future.
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