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 e 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 e, while capturing the correct Kerr-Lorentz limit as e → 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 e in the model is also examined numerically.

[1]  Giacomo Dimarco,et al.  Exponential Runge-Kutta Methods for Stiff Kinetic Equations , 2010, SIAM J. Numer. Anal..

[2]  Yan Jiang,et al.  Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media , 2017, J. Comput. Phys..

[3]  Bernardo Cockburn,et al.  Interpolatory HDG Method for Parabolic Semilinear PDEs , 2018, Journal of Scientific Computing.

[4]  Richard W. Ziolkowski,et al.  Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time , 1993 .

[5]  Daniel A. White,et al.  An efficient vector finite element method for nonlinear electromagnetic modeling , 2007, J. Comput. Phys..

[6]  Jingwei Hu,et al.  A Second-Order Asymptotic-Preserving and Positivity-Preserving Exponential Runge-Kutta Method for a Class of Stiff Kinetic Equations , 2018, Multiscale Model. Simul..

[7]  D. Aregba-Driollet.  Godunov scheme for Maxwell's equations with Kerr nonlinearity , 2014, 1404.6372.

[8]  P Kinsler,et al.  Optical carrier wave shocking: detection and dispersion. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Tong Wu,et al.  Steady State and Sign Preserving Semi-Implicit Runge-Kutta Methods for ODEs with Stiff Damping Term , 2015, SIAM J. Numer. Anal..

[10]  P. Degond.  Asymptotic-Preserving Schemes for Fluid Models of Plasmas , 2011, 1104.1869.

[11]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[12]  J V Moloney,et al.  Electromagnetic shocks on the optical cycle of ultrashort pulses in triple-resonance Lorentz dielectric media with subfemtosecond nonlinear electronic Debye relaxation. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  William L. Kath,et al.  Numerical solutions of Maxwell’s equations for nonlinear-optical pulse propagation , 1996 .

[14]  Shi Jin.  ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW , 2010 .

[15]  Zhichao Peng,et al.  Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling , 2020, J. Comput. Phys..

[16]  A. G. Llnl Taylor.  User documentation for KINSOL, a nonlinear solver for sequential and parallel computers , 1998 .

[17]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[18]  Susan C. Hagness,et al.  Comparison between Staggered and Unstaggered Finite-Difference Time-Domain Grids for Few-Cycle Temporal Optical Soliton Propagation , 2000 .

[19]  Yan Jiang,et al.  High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension , 2018, J. Sci. Comput..

[20]  Xiangxiong Zhang,et al.  Asymptotic-Preserving and Positivity-Preserving Implicit-Explicit Schemes for the Stiff BGK Equation , 2017, SIAM J. Numer. Anal..

[21]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[22]  A. D. L. Bourdonnaye.  High-Order Scheme for a Nonlinear Maxwell System Modelling Kerr Effect , 2000 .

[23]  Fabrice Deluzet,et al.  Asymptotic-Preserving methods and multiscale models for plasma physics , 2016, J. Comput. Phys..

[24]  Robert A. Fisher,et al.  Introduction to Nonlinear Optics , 2006 .

[25]  Juntao Huang,et al.  A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model , 2017 .

[26]  Nicolas Crouseilles,et al.  A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling , 2017 .

[27]  Jim Douglas,et al.  The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures , 1975 .

[28]  Nicolaas Bloembergen,et al.  Nonlinear Optics , 1977 .