177 research outputs found
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension
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.Comment: 47 pages, 8 figure
A High-Order Numerical Method for the Nonlinear Helmholtz Equation in Multidimensional Layered Media
We present a novel computational methodology for solving the scalar nonlinear
Helmholtz equation (NLH) that governs the propagation of laser light in Kerr
dielectrics. The methodology addresses two well-known challenges in nonlinear
optics: Singular behavior of solutions when the scattering in the medium is
assumed predominantly forward (paraxial regime), and the presence of
discontinuities in the % linear and nonlinear optical properties of the medium.
Specifically, we consider a slab of nonlinear material which may be grated in
the direction of propagation and which is immersed in a linear medium as a
whole. The key components of the methodology are a semi-compact high-order
finite-difference scheme that maintains accuracy across the discontinuities and
enables sub-wavelength resolution on large domains at a tolerable cost, a
nonlocal two-way artificial boundary condition (ABC) that simultaneously
facilitates the reflectionless propagation of the outgoing waves and forward
propagation of the given incoming waves, and a nonlinear solver based on
Newton's method.
The proposed methodology combines and substantially extends the capabilities
of our previous techniques built for 1Dand for multi-D. It facilitates a direct
numerical study of nonparaxial propagation and goes well beyond the approaches
in the literature based on the "augmented" paraxial models. In particular, it
provides the first ever evidence that the singularity of the solution indeed
disappears in the scalar NLH model that includes the nonparaxial effects. It
also enables simulation of the wavelength-width spatial solitons, as well as of
the counter-propagating solitons.Comment: 40 pages, 10 figure
Global Artificial Boundary Conditions for Computation of External Flow Problems with Propulsive Jets
We propose new global artificial boundary conditions (ABC's) for computation of flows with propulsive jets. The algorithm is based on application of the difference potentials method (DPM). Previously, similar boundary conditions have been implemented for calculation of external compressible viscous flows around finite bodies. The proposed modification substantially extends the applicability range of the DPM-based algorithm. In the paper, we present the general formulation of the problem, describe our numerical methodology, and discuss the corresponding computational results. The particular configuration that we analyze is a slender three-dimensional body with boat-tail geometry and supersonic jet exhaust in a subsonic external flow under zero angle of attack. Similarly to the results obtained earlier for the flows around airfoils and wings, current results for the jet flow case corroborate the superiority of the DPM-based ABC's over standard local methodologies from the standpoints of accuracy, overall numerical performance, and robustness
Simulations of the Nonlinear Helmholtz Equation: Arrest of Beam Collapse, Nonparaxial Solitons, and Counter-Propagating Beams
We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that
collapse in the simpler NLS model. Thereby, we provide the first ever numerical
evidence that nonparaxiality and backscattering can arrest the collapse. We
also solve the (1+1)D NLH and show that solitons with radius of only half the
wavelength can propagate over forty diffraction lengths with no distortions. In
both cases we calculate the backscattered field, which has not been done
previously. Finally, we compute the dynamics of counter-propagating solitons
using the NLH model, which is more comprehensive than the previously used
coupled NLS model.Comment: 6 pages, 6 figures, Lette
Transparent boundary conditions based on the Pole Condition for time-dependent, two-dimensional problems
The pole condition approach for deriving transparent boundary conditions is
extended to the time-dependent, two-dimensional case. Non-physical modes of the
solution are identified by the position of poles of the solution's spatial
Laplace transform in the complex plane. By requiring the Laplace transform to
be analytic on some problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of
a series expansion of the Laplace transform, thereby providing an approximation
to the exact boundary condition. The resulting error decays super-algebraically
with the number of coefficients, so relatively few additional degrees of
freedom are sufficient to reduce the error to the level of the discretization
error in the interior of the computational domain. The approach shows good
results for the Schr\"odinger and the drift-diffusion equation but, in contrast
to the one-dimensional case, exhibits instabilities for the wave and
Klein-Gordon equation. Numerical examples are shown that demonstrate the good
performance in the former and the instabilities in the latter case
Adaptive absorbing boundary conditions for Schrodinger-type equations: application to nonlinear and multi-dimensional problems
We propose an adaptive approach in picking the wave-number parameter of
absorbing boundary conditions for Schr\"{o}dinger-type equations. Based on the
Gabor transform which captures local frequency information in the vicinity of
artificial boundaries, the parameter is determined by an energy-weighted method
and yields a quasi-optimal absorbing boundary conditions. It is shown that this
approach can minimize reflected waves even when the wave function is composed
of waves with different group velocities. We also extend the split local
absorbing boundary (SLAB) method [Z. Xu and H. Han, {\it Phys. Rev. E},
74(2006), pp. 037704] to problems in multidimensional nonlinear cases by
coupling the adaptive approach. Numerical examples of nonlinear Schr\"{o}dinger
equations in one- and two dimensions are presented to demonstrate the
properties of the discussed absorbing boundary conditions.Comment: 18 pages; 12 figures. A short movie for the 2D NLS equation with
absorbing boundary conditions can be downloaded at
http://home.ustc.edu.cn/~xuzl/movie.avi. To appear in Journal of
Computational Physic
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