xi, 172 p. : ill. (some col.) / Accretion shock waves are present in many important astrophysical systems and have been a focus of research for decades. These investigations provide a large body of understanding as to the nature, characteristics, and evolutionary behaviors of accretion shock waves over a wide range of conditions. However, largely absent are investigations into the properties of accretion shock waves in the presence of strong magnetic fields. In such cases these strong magnetic fields can significantly alter the stability behaviors and evolution of the accretion shock wave through the production and propagation of magnetic waves as well as magnetically constrained advection. With strong magnetic fields likely found in a number of accretion shock systems, such as compact binary and protostellar systems, a better understanding of the behaviors of magnetic accretion shock waves is needed. A new magnetohydrodynamics simulation tool, IMOGEN, was developed to carry out an investigation of instabilities in strong, slow magnetic accretion shocks by modelling their long-term, nonlinear evolution. IMOGEN implements a relaxed, second-order, total variation diminishing, monotonic upwind scheme for conservation laws and incorporates a staggered-grid constrained transport scheme for magnetic advection. Through the simulated evolution of magnetic accretion shocks over a wide range of initial conditions, it has been shown, for sufficiently high magnetic field strengths, that magnetic accretion shocks are generally susceptible to corrugation instabilities, which arise in the presence of perturbations of the initial shock front. As these corrugation instabilities grow, they manifest as magnetic wave propagation in the upstream region of the accretion column, which propagate away from the accretion shock front, and as density columns, or fingers, that grow into the higher density downstream flow, defined and constrained by current loops created during the early evolution of the instability. / Committee in charge: Dr. James Schombert, Chair; Dr. James Imamura, Advisor; Dr. Alan Rempel, Member; Dr. John Toner, Member; Dr. Kent Stevens, Outside Member