Asymptotic and positivity preserving methods for Kerr-Debye model with Lorentz dispersion in one dimension
Asymptotic and positivity preserving methods for Kerr-Debye model with Lorentz dispersion in one dimension
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DOI：
10.1016/j.jcp.2019.109101
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
Zhichao Peng;V. Bokil;Yingda Cheng;Fengyan Li
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文献类型：
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作者：
Zhichao Peng;V. Bokil;Yingda Cheng;Fengyan Li
在本文中，我们继续我们在[4]，[5]中的最新发展，设计具有重要可证明性质的数值方法来模拟非线性光学介质中的电磁波传播。特别地，我们考虑了具有Lorentz色散的一维Kerr-Debye模型，称为Kerr-Debye-Lorentz模型，其中偏振的非线性是松弛的立方Kerr型效应。偏振还包括线性洛伦兹色散。随着弛豫时间ε趋于零，模型将接近Kerr-Lorentz模型。本文的目的是设计和分析Kerr-Debye-Lorentz模型的保渐近（AP）和保正性（PP）方法。作为AP，这些方法解决了与小ε相关的模型刚度，同时在欠分辨网格上捕获正确的Kerr-Lorentz极限ε → 0。由于是PP，三阶非线性极化率将保持非负，这对能量稳定性很重要。在所提出的方法中，节点不连续的Galerkin（DG）离散的任意阶精度在空间上应用，以有效地处理非线性，在时间上，几个一阶和二阶方法的开发。我们证明了一阶时间全离散格式是AP、PP格式，并且是能量稳定的。对于二阶时间精度，提出了一种新的修改的指数时间积分器的刚性部分的辅助微分方程建模的电极化，这是一个关键成分的方法是AP和PP。除了一个简单的离散的本构关系，我们进一步提出了一个非平凡的基于能量的近似，与能量稳定性也建立数学。数值算例包括一个ODE的例子，一个制造的解决方案，类孤子的传播和Sech信号在熔融石英块体的传播，比较所提出的方法，并证明的准确性，AP和PP属性。在模型中的有限弛豫时间ε的影响也进行了数值研究。
In this paper, we continue our recent developments in [4],[5] to devise numerical methods that have important provable properties to simulate electromagnetic wave propagation in nonlinear optical media. Particularly, we consider the one dimensional Kerr-Debye model with the Lorentz dispersion, termed as the Kerr-Debye-Lorentz model, where the nonlinearity in the polarization is a relaxed cubic Kerr type effect. The polarization also includes the linear Lorentz dispersion. As the relaxation time ε goes to zero, the model will approach the Kerr-Lorentz model. The objective of this work is to devise and analyze asymptotic preserving (AP) and positivity preserving (PP) methods for the Kerr-Debye-Lorentz model. Being AP, the methods address the stiffness of the model associated with small ε, while capturing the correct Kerr-Lorentz limit as ε→ 0 on under-resolved meshes. Being PP, the third-order nonlinear susceptibility will stay non-negative and this is important for the energy stability. In the proposed methods, the nodal discontinuous Galerkin (DG) discretizations of arbitrary order accuracy are applied in space to effectively handle nonlinearity; in time, several first and second order methods are developed. We prove that the first order in time fully discrete schemes are AP, PP and also energy stable. For the second order temporal accuracy, a novel modified exponential time integrator is proposed for the stiff part of the auxiliary differential equations modeling the electric polarization, and this is a key ingredient for the methods to be both AP and PP. In addition to a straightforward discretization of the constitutive law, we further propose a non-trivial energy-based approximation, with which the energy stability is also established mathematically. Numerical examples are presented that include an ODE example, a manufactured solution, the soliton-like propagation and the propagation of Sech signal in fused bulk silica, to compare the proposed methods and to demonstrate the accuracy, AP and PP property. The effect of the finite relaxation time ε in the model is also examined numerically.