Computational "non-local" electrodynamics (2020)
The classical theory of electrodynamics, represented by differential operators, describes the "local" dependence between electric and magnetic fields at every location in spacetime. Using the FDTD technique, the discretized version of Maxwell's equations uses central difference approximations for the numerical derivatives in space and time. This dissertation thesis presents a novel time-symmetrical "non-local" technique to solve the wave equation through the formalism of the electromagnetic potential. When the symmetric contribution over time is identified from the wave equation, the resulting formulation is more straightforward, time-symmetric, and "non-local". An algorithm was proposed in which the computational domain is tessellated in the form of diamonds where the potentials are located along characteristic surfaces. From them, the solutions on other characteristic surfaces are computed "non-locally". In the proposed case study, the "non-local" proposal is 1643 faster than FDTD, that is, more than three orders of magnitude, and uses a time-step that is 4096 times greater than the Courant-Friedrichs-Lewy limit without encountering instability problems. The performance gain is proportional to the size of the spacetime tessellations since the "domain-to-boundary" ratio increases when the domain's extension does. Consequently, this novel "non-local" approach reduces computational complexity, and it provides a more comprehensive explanation of its fundamental physical aspects, without contradicting the principles of the successful classical and "local" field theory. Based on this time-symmetrical "non-local" approach, a formulation of the mean value theorem for time-varying electromagnetic potentials applied to electrodynamics is proposed