Wave
propagation in inhomogeneous media
In nature, electromagnetic waves propagate both in free space and
in bodies which may be inhomogeneous media. For instance, a
cellular phone sends signals from a building to the closest antenna
to register its location. Another example is a sensor that emits
electromagnetic pulses into the ground to check for land mines.
These can be simulated by the solution of Maxwell equations with
discontinuous coefficients. A discontinuity in coefficients
represents the propagation of electromagnetic waves between media
with different dielectric and magnetic properties.
In our work we use an approach called “regularization” that enables
to smooth discontinuous coefficients and to solve Maxwell equations
with minimal lost in accuracy of numerical scheme. This approach is
tested on grid conforming dielectric bodies in both frequency
domain (Helmholtz and Maxwell equations) and time domain (Maxwell
equations) using numerical and asymptotic techniques for error
analysis. Now we are developing regularization algorithms for high
order accurate study of electromagnetic wave propagation through
general dielectric structures in two and three dimensions.
Publications:
E. Kashdan, E Turkel, "A High-Order Accurate Method for the
Frequency Domain Maxwell’s Equations with Discontinuous
Coefficients", Journal of Scientific Computing, 27(1-3), 2006,
pp. 75—95.
E. Kashdan, E Turkel, "High Order Accurate Modelling of
Electromagnetic Wave Propagation Across Media — Grid Conforming
Bodies", Journal of Computational Physics, 218(2), 2006, pp.
816-835
Artificial boundary conditions
Numerical modeling of wave propagation on unbounded domains
requires the proper boundary conditions to stop outgoing waves
coming back and disrupting the computations. Since its introduction
by Berenger in 1994 the Perfectly Matched Layers (PML) has become a
most popular approach for non-reflecting Artificial Boundary
Conditions (ABC) in electromagnetics. However, all popular PML
algorithms double the number of equations to be solved inside the
artificial domain in Cartesian coordinates in 3D. In some cases the
implementation of the PML leads to temporal growth of the
reflections into the physical domain or (and) instabilities.
We have developed and implemented a new concept of ABC – nonlinear
PML (NLPML). The system of NLPML equations is strongly well posed
as an initial value problem; a monochromatic plane wave propagating
in a given direction, upon entering the PML, will decay (in all
directions) spatially; the energy integral of the solution is
temporally stable and the total number of equations inside the PML
does not exceed the number of Maxwell equations (6 in 3D). The
numerical tests show that the NLPML outperforms all other models of
PML (e.g. has less non-physical reflections into the computational
domain) in the long time integration and on the problems with the
dielectric
interfaces.
In our recent work, we applied the concept of NLPML to advective
acoustics (Linearized Euler equations – LEE). The first numerical
results show that the NLPML outperforms all other types of PML in
2D, especially for small number of layers, which means that we need
much less computer memory and time to get the results within
required accuracy limits. Our NLPML model works also in 3D and it
is effectively the only tested three-dimensional model of ABC for
LEE, – most of the existing ABC have no 3D formulations.
Publications:
S. Abarbanel, E. Kashdan, “Nonlinear Perfectly Matched Layers
for Advective Acoustics”, submitted 2010
S. Abarbanel, E. Kashdan, "Nonlinear PML equations and their
embedding into the FDTD framework", 6th Workshop on
Computational Electromagnetics in Time-Domain, pp. 92 – 95,
Atlanta, USA, 2005.
Splitting methods
The dispersal and mixing of scalar quantities such as
concentrations or thermal energy are often modeled by
advection-diffusion equations. Such problems arise in a wide
variety of engineering, ecological and geophysical applications. In
these situations a quantity such as chemical or pollutant
concentration or temperature variation diffuses while being
transported by the governing flow. Both steady laminar and complex
(chaotic, turbulent or random) time-dependent flows are of
interest. The development of reliable numerical methods for
advection-diffusion equations is crucial for understanding their
properties, both physical and mathematical.
The idea behind the splitting method is to use different schemes
for the advection and diffusion computations. In our work we extend
the fast explicit operator
splitting method to problems with random coefficients and singular
sources.
Publications:
A. Chertock, C. R. Doering, E. Kashdan and A. Kurganov, “A Fast
Explicit Operator Splitting method for passive scalar advection by
a random stirring field”, accepted for publication in Journal
of Scientific Computing, 2010.
A. Chertock, E. Kashdan and A. Kurganov, “Propagation of
Diffusing Pollutant by a Hybrid Eulerian-Lagrangian Method”,
in Hyperbolic Problems: Theory, Numerics, Applications, S.
Benzioni-Gavage and D. Serre, eds., Springer, 2008, pp.
371-380.
Parallel algorithms
In this project we implement fourth-order accurate compact implicit
scheme for solution of time-dependent Maxwell equations. This
scheme requires additional memory resources and increases
computation time compare to the widely used second order accurate
FDTD algorithm. However, symmetry and linearity of the system of
Maxwell equations allows the construction of effective parallel
algorithm based on alternating domain decomposition that
significantly improves performance of fourth order scheme due to
the minimization of inter-processor communication time.
Publications:
E. Kashdan, B. Galanti, "A new parallelization strategy for
solving time-dependent 3D Maxwell's equations using a high-order
accurate compact implicit scheme", International Journal of
Numerical Modeling, 19(5), 2006, pp. 391—408.