Neutron stars and black holes are generally surrounded by magnetospheres of highly conducting plasma in which the magnetic flux density is so high that hydrodynamic forces are irrelevant. In this vanishing-inertia---or ultra-relativistic---limit, magnetohydrodynamics becomes force-free electrodynamics, a system of equations comprising only the magnetic and electric fields, and in which the plasma response is effected by a nonlinear current density term. In this dissertation I describe a new pseudospectral simulation code, designed for studying the dynamic magnetospheres of compact objects. A detailed description of the code and several numerical test problems are given. I first apply the code to the aligned rotator problem, in which a star with a dipole magnetic field is set rotating about its magnetic axis. The solution evolves to a steady state, which is nearly ideal and dissipationless everywhere except in a current sheet, or magnetic field discontinuity, at the equator, into which electromagnetic energy flows and is dissipated.
Magnetars are believed to have twisted magnetospheres, due to internal magnetic evolution which deforms the crust, dragging the footpoints of external magnetic field lines. This twisting may be able to explain both magnetars' persistent hard X-ray emission and their energetic bursts and flares. Using the new code, I simulate the evolution of relativistic magnetospheres subjected to slow twisting through large angles. The field lines expand outward, forming a strong current layer; eventually the configuration loses equilibrium and a dynamic rearrangement occurs, involving large-scale rapid magnetic reconnection and dissipation of the free energy of the twisted magnetic field. When the star is rotating, the magnetospheric twisting leads to a large increase in the stellar spin-down rate, which may take place on the long twisting timescale or in brief explosive events, depending on where the twisting is applied and the history of the system. One such explosive field-expansion and reconnection event may have been responsible for the 27 August 1998 giant flare from SGR 1900+14, and the coincident sudden increase in spin period, or "braking glitch." The inner magnetospheres of relativistic compact objects are in strongly curved spacetimes.
I describe the extension of the code to general-relativistic simulations, including the hypersurface foliation method and the 3+1 equations of force-free electrodynamics in curved, evolving spacetimes. A simple test problem for dynamical behavior in the Schwarzschild metric is presented, and the evolutions of the magnetospheres surrounding neutron stars and black holes, in vacuum and in force-free plasma, are compared.
