In this thesis we construct compatible discretizations of Maxwell’s equations. We use the term compatible to describe numerical methods for Maxwell’s equations which obey many properties of vector Calculus in a discrete setting. Compatible discretizations preserve the exterior Calculus ensuring that the divergence of the curl and the curl of a gradient are zero in a discrete setting. This compatibility of discretizations with the continuum Maxwell’s equations guarantees that the numerical solutions are physically meaningful.
We focus on the construction of a class of discretizations called Mimetic Finite Differences (MFD). The MFD method is a generalization of both staggered finite differences and mixed finite elements. We construct a parameterized family of MFD methods with equivalent formal order of accuracy. For time-dependent problems, we exploit this non-uniqueness by finding parameters which are optimal with respect to a certain criteria, for example, minimizing dispersion error. Dispersion error is a numerical artifact in which individual frequencies in a wave propagate at incorrect speeds; dominating the error in wave problems over long time propagation. The novelty of this work is the construction of an MFD discretization for Maxwell’s equations which reduces dispersion error for transient wave propagation in materials that are modeled by a general class of linear constitutive laws. We provide theoretical analysis of these new discretizations including an analysis of stability and discrete divergence. We also provide numerical demonstrations to illustrate the theory.
In addition to applications in the time domain we consider equilibrium Magnetohydrodynamic (MHD) generators. MHD generators extract power directly from a plasma by passing it through a strong magnetic field. Used as a topping cycle for traditional steam turbine generator, MHD offers a theoretical thermal efficiency of 60% compared to 40% of traditional systems. However, this technology has high life cycle costs due to equipment failure. One source of failure is arcing: the formation of high density currents which damage the generator. In this work we develop, analyze, and simulate a model of these generators. We use these simulations to show the viability of detecting electrical arcs by measurements of their magnetic fields outside of the generator.