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.