Encoding the electrodynamics in spatiotemporal boundaries

Abstract

Maxwell’s equations represented by differential operators describe the local dependence between both electric and magnetic fields in every location of space and time. This description responds to the field concept proposed by Michael Faraday and formalized by James Clerk Maxwell. The discretized version in the form of the finite-difference time-domain (FDTD) technique relies on a proposal of local computation of the respective differential operators making use of the central difference approximation of the second derivative of functions through the Taylor series expansion. This article introduces a novel time-symmetric “non-local” technique from the mathematical formalism of electromagnetic potentials in the wave equation and its physical interpretation in the Minkowski spacetime. In the proposed case study, the “non-local” proposal is 1643 faster than FDTD, i.e., more than three orders of magnitude, and it uses a time-step that is 4096 times greater than the Courant–Friedrichs–Lewy limit without encountering stability problems. New electromagnetic potentials are calculated from the previous and distant ones located at vertices of spatiotemporal regions called causal diamonds that tessellates spacetime without the need for computations inside them. We show that the performance gain is proportional to the size of the spacetime tessellations because the ratio “domain to boundary” increases as the domain extension does. Consequently, our novel “non-local” approach provides a reduction in computational complexity and a more comprehensible explanation of their fundamental physical aspects, without contradicting the principles of the successful classical field theory. Program summaryProgram Title: SOME_modelsProgram Files doi:http://dx.doi.org/10.17632/7sgh6grm7g.1Licensing provisions: GPLv3Programming language: CNature of problem: The Courant–Friedrichs–Lewy (CFL) condition imposes a numerical bound to the size of the Yee cell in the FDTD approach. This restriction causes excessive oversampling, degrading the computation time of this technique.Solution method: We applied the parallelogram rule to solve the wave equation for the case of (1+1) time–space dimensions. With this rule, we proposed a novel algorithm that allows encoding potentials along the characteristic lines that tessellate the spacetime. The new technique produces higher performance gains than those achieved through traditional FDTD.