An extension of the finite difference time domain is applied to solve the Schrödinger equation. A systematic analysis of stability and convergence of this technique is carried out in this article. The numerical scheme used to solve the Schrödinger equation differs from the scheme found in electromagnetics. Also, the unit cell employed to model quantum devices is different from the Yee cell used by the electrical engineering community. A bound for the time step is derived to ensure stability. Several numerical experiments in quantum structures demonstrate the accuracy of a second order, comparable to the analysis of electromagnetic devices with the Yee cell.

[1] A numerical modeling of Earth's atmosphere is carried out by means of the Transmission Line Matrix (TLM) numerical method with the aim of calculating the Schumann resonance frequencies. The numerical results obtained are very close to the experimental ones and those obtained with the widely accepted two-scale-height ionospheric model, which allows us to affirm that this is a valid numerical tool for predicting the Schumann frequencies in the atmospheres of other planets and moons. The great flexibility of the TLM numerical method also allows the study of slight shifts in the Schumann resonance frequencies due to an increase in electrical conductivity at the Earth's poles, originating from high-energy particle precipitation emitted from the Sun in conjunction with solar flares. A slight increase in the first Schumann resonance frequency is observed during these events, which is associated with a reduction in the dimensions of the electromagnetic cavity.

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This paper presents a numerical approach to study the electrical properties of the Earth's atmosphere. The finite-difference time-domain (FDTD) technique is applied to model the Earth's atmosphere in order to determine Schumann resonant frequencies of the Earth. Three-dimensional spherical coordinates are employed and the conductivity profile of the atmosphere versus height is introduced. Periodic boundary conditions are implemented in order to exploit the symmetry in rotation of the Earth and decrease computational requirements dramatically. For the first time, very accurate FDTD results are obtained, not only for the fundamental mode but also for higher order modes of Schumann resonances. The proposed method constitutes a useful tool to obtain Schumann resonant frequencies, therefore to validate electrical models for the terrestrial atmosphere, or atmospheres of other celestial bodies.Index Terms-Earth-ionosphere waveguide, extremely low frequency (ELF), finite-difference time-domain (FDTD) methods, propagation.

[1] The study of the propagation of extremely low frequency (ELF) waves is essential for electromagnetic sounding investigations planned for some of the future Martian missions. Future surface stations will have the possibility of continuously recording low-frequency electromagnetic field fluctuations. Natural electromagnetic waves produced near the surface by electrostatic discharges in dust storms (dust devils) or by geological activity can be trapped in the resonant cavity formed by the surface and lower ionosphere as it occurs on the Earth. Low-frequency electromagnetic waves can also travel along the magnetic field lines of the recently discovered magnetic anomalies from the magnetosphere to the surface and may produce resonant structures in the cavity. The structure of the resonant frequencies, also called Schumann frequencies, is mainly determined by the geometry of the cavity and by the global electrical conductivity of the ionosphere/atmosphere. Measurements of Schumann frequencies by surface stations can be used for remote sensing of the electrical conductivity of the lower ionosphere/ atmosphere. We present a numerical model of electromagnetic wave propagation based on the transmission line modeling (TLM) method with the aim of calculating the resonance frequencies on Mars and their dependence on solar activity and various possible ionization sources like meteoroids. The model has been previously validated by application to the terrestrial case. The numerical results obtained for the Earth are very close to the experimental ones, which supports our predictions on Mars. Our model can be used to study the global atmospheric conductivity using future real ELF measurements by surface stations or even balloons on Mars.Citation: Molina-Cuberos, G. J., J. A. Morente, B. P. Besser, J. Portí, H. Lichtenegger, K. Schwingenschuh, A. Salinas, and J. Margineda (2006), Schumann resonances as a tool to study the lower ionospheric structure of Mars, Radio Sci., 41, RS1003,