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Joint Applied Math and Probability Seminar
The Huygens' Principle in Computational Electromagnetics and Beyond
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If the waves governed by a particular hyperbolic PDE (or system) have sharp aft fronts, we say that the differential equation satisfies the Huygens' principle. The areas of no disturbance behind the aft fronts are called lacunae; their existence is a rather delicate and fragile property. Otherwise, the propagation of waves is accompanied by after- effects often referred to as the wave diffusion. Lacunae can be effectively exploited for the numerical simulation of unsteady waves. Even though the diffusionless propagation of waves is generally rare, it can still be observed in a number of important applications, including the Maxwell equations. In the talk, we will review some recent progress on the development and implementation of lacunae-based methods in computational electromagnetics. Existence of the lacunae for the Maxwell system is still not automatic. The continuity requirement for the charges and currents may present a serious obstacle. To overcome it, a special inverse problem is to be formulated and solved of reconstructing the auxiliary field sources in a prescribed form that satisfies the continuity equation identically. Its relation to the classical inverse scattering and inverse source problems, and the problem of non-radiating currents, will be explored. Maxwell's equations is a first order system that reduces to a set of wave equations; that's why it satisfies the Huygens' principle. In the conventional 3+1 dimensional Minkowski space-time the only scalar Huygens' equation happens to be the wave equation. There are, however, systems that do not reduce to the wave equation and are still Huygens'; there are also scalar irreducible Huygens' examples in higher dimension Minkowski spaces, as well as in the 3+1-dimensional space with alternative metrics. Some of those examples will be discussed along with the (remote) possibility of assigning a tangible physical interpretation to any of them. On the other hand, some simple yet useful physical models for metals, semiconductors, dielectrics, and plasmas will be discussed from the standpoint of whether lacunae may exist in the corresponding solutions and if not, whether the aft fronts can still be identified in some approximate sense. |