Quasi-Lacunae of Maxwell's Equations

Abstract

Classical lacunae in the solutions of hyperbolic differential equations and systems (in the spaces of odd dimension) are a manifestation of the Huygens' principle. If the source terms are compactly supported in space and time, then, at any finite location in space, the solution becomes identically zero after a finite interval of time. In other words, the propagating waves have sharp aft fronts. For Maxwell's equations though, even if the currents that drive the field are compactly supported in time, they may still lead to the accumulation of charges. In that case, the solution won't have the lacunae per se. We show, however, that the notion of classical lacunae can be generalized, and that even when the steady-state charges are present, the waves still have sharp aft fronts. Yet behind those aft fronts, there is a nonzero electrostatic solution rather than one identically zero.

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

“…Two types of methods that have been introduced and analyzed previously can prove useful in this context: the methods for unsteady control of sound [59], and lacunae-based methods that are used for setting the unsteady ABCs [44,56,58], as well as for achieving the improved performance over long times [35][36][37].…”

“…It approximates the governing differential equation on Ω from the definition of the original boundary value problem (24). The key property of the projection P γ in (34) parallels the corresponding key property in the continuous case, see (22a), (22b),-a given ξ γ satisfies the inhomogeneous difference BEP: (35) holds, then the corresponding solution u on N + is given by the discrete generalized Green's formula [cf. formula (23)]:…”

“…As indicated in the end of Sect. 3.2, the source term f (x) for the boundary value problem (24) may not even be defined anywhere beyond the domain Ω, but the discrete right-hand side f needs to be known at the nodes K + \ K Ω in order to apply the operator B (h) in formulae (35) and (36). To obtain f at the nodes K + \ K Ω , which are located close to Γ yet outside Ω, we will also employ the Taylor formula [cf.…”

“…4.2 and in Appendix A. We will now seek ξ ξ ξ Γ in the form (40b), for which the extension (48), or more generally, (87), satisfies the discrete BEP (35).…”

“…Once c is known, the extension (87) of the resulting ξ ξ ξ Γ of (40b) will satisfy the discrete BEP (35). Since, the BEP is equivalent only to the governing equation on the domain and does not account for the boundary conditions, then, in addition to system (53), one still needs to consider the boundary condition of (24) in order to uniquely determine c.…”