Conservative modeling of 3-D electromagnetic fields, Part I: Properties and error analysis

Abstract

Conservation of electric current and magnetic flux can be explicitly enforced by modeling Maxwell’s equations on a staggered grid, where the different field components are sampled at points offset relative to each other. A staggered finite‐difference (SFD) approximation gives divergence‐free magnetic fields and electric currents ensuring good behavior at all periods. Comparisons of SFD solutions with 2-D quasi‐analytic solutions are very good (∼1% rms error). When a modeled region can be subdivided into uniform subdomains, comparison of analytic solutions and SFD approximations show that the greatest differences occur near the Nyquist wavenumbers; the SFD solutions do not attenuate in space as rapidly as the analytic solutions. The accuracy of a computed SFD solution can be estimated from its wavenumber content. For test cases the accuracy estimates are surprisingly close to the actual accuracies. Grid requirements for modeling short horizontal wavelength components of a solution seem more demanding than for modeling the infinite wavelength (plane‐wave) component: for 2% accuracy 4π samples are required per horizontal wavelength compared to two samples per skin depth for the plane‐wave component.