Asymptotic and positivity preserving methods for Kerr-Debye model with Lorentz dispersion in one dimension

Abstract

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.