High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

Details High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

2007-08-06

arxiv.org/abs/0708.0726v2

Journal of Computational Physics, 227 1 (Nov. 2007) pp. 820-850

Physics

Mathematical Physics

47 pages, 8 figures

Scientific paper

10.1016/j.jcp.2007.08.022

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.

Affiliated with

Also associated with

No associations

Rating

High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-403804