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

G. Baruch, G. Fibich, S. Tsynkov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


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. Our computations corroborate the design fourth order convergence of the method. The one-dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future.

Original languageEnglish
Pages (from-to)820-850
Number of pages31
JournalJournal of Computational Physics
Issue number1
StatePublished - 10 Nov 2007


FundersFunder number
National Science Foundation
Directorate for Mathematical and Physical Sciences0509695
Directorate for Mathematical and Physical Sciences


    • Artificial boundary conditions (ABCs)
    • Compact scheme
    • Complex valued solutions
    • Discontinuous coefficients
    • Finite volume discretization
    • Frechét differentiability
    • High-order method
    • Inhomogeneous medium
    • Kerr nonlinearity
    • Newton's method
    • Nonlinear optics
    • Traveling waves
    • Two-way ABCs


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