4,379 research outputs found
A new integral representation for quasiperiodic fields and its application to two-dimensional band structure calculations
In this paper, we consider band-structure calculations governed by the
Helmholtz or Maxwell equations in piecewise homogeneous periodic materials.
Methods based on boundary integral equations are natural in this context, since
they discretize the interface alone and can achieve high order accuracy in
complicated geometries. In order to handle the quasi-periodic conditions which
are imposed on the unit cell, the free-space Green's function is typically
replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic
Green's function diverges for families of parameter values that correspond to
resonances of the empty unit cell. Here, we bypass this problem by means of a
new integral representation that relies on the free-space Green's function
alone, adding auxiliary layer potentials on the boundary of the unit cell
itself. An important aspect of our method is that by carefully including a few
neighboring images, the densities may be kept smooth and convergence rapid.
This framework results in an integral equation of the second kind, avoids
spurious resonances, and achieves spectral accuracy. Because of our image
structure, inclusions which intersect the unit cell walls may be handled easily
and automatically. Our approach is compatible with fast-multipole acceleration,
generalizes easily to three dimensions, and avoids the complication of
divergent lattice sums.Comment: 25 pages, 6 figures, submitted to J. Comput. Phy
High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods
Arbitrary high order numerical methods for time-harmonic acoustic scattering
problems originally defined on unbounded domains are constructed. This is done
by coupling recently developed high order local absorbing boundary conditions
(ABCs) with finite difference methods for the Helmholtz equation. These ABCs
are based on exact representations of the outgoing waves by means of farfield
expansions. The finite difference methods, which are constructed from a
deferred-correction (DC) technique, approximate the Helmholtz equation and the
ABCs, with the appropriate number of terms, to any desired order. As a result,
high order numerical methods with an overall order of convergence equal to the
order of the DC schemes are obtained. A detailed construction of these DC
finite difference schemes is presented. Additionally, a rigorous proof of the
consistency of the DC schemes with the Helmholtz equation and the ABCs in polar
coordinates is also given. The results of several numerical experiments
corroborate the high order convergence of the novel method.Comment: 36 pages, 20 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
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
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