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Center for Nonlinear Studies |
ABSTRACT: We describe polarization models for polydispersive media, such as biological tissue. One popular approach for representing the polarization is the Cole-Cole (1936) model, a heuristic generalization of the standard Debye (1929) model which itself corresponds to a first order linear auxiliary ODE. The Cole-Cole model corresponds to a fractional order ODE which presents computational challenges. We describe an alternative approach based on using the Debye model, but with a probability distribution of relaxation times. We introduce a novel numerical approach based on Polynomial Chaos Expansions. We then consider stability and dispersion analysis of staggered, finite difference time domain (FDTD) methods for Maxwell's equations in these polydispersive media. In particular, we present a proof of (conditional) stability using an energy decay property and derive a discrete dispersion relation in terms of a discrete expected complex permitivity. Lastly we describe the extension to high order accurate (in space) FDTD methods.