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The primary growth pathway of both stellar mass black holes (such as those found in X-ray binary systems), and supermassive black holes (found in active galactic nuclei) is through the accretion of gas. These systems are some of the most energetic in the Universe and provide the opportunity to explore fundamental physics in regimes that are unattainable here on Earth. However, utilizing these unique laboratories to their full potential is difficult because the physics driving the accretion of matter is poorly understood. In this talk I will discuss recent modeling efforts aimed at elucidating the physics behind accretion and, in particular, the variability observed from these systems. My primary tools for this work are large, global simulations of the turbulent flows within the accretion disks. I will present hydrodynamic and magnetohydrodynamic simulations of the “truncated” accretion disk, which is believed to arise in the inner region of an accretion disk when the gas reaches high temperatures and becomes radiatively inefficient. I will also discuss a magnetohydrodynamic simulation that shows the launching of “propagating fluctuations” in mass accretion rate, which we find are driven by the magnetic dynamo of the disk. These propagating fluctuations are thought to explain the unique nonlinear signatures found the in the photometric variability from accreting black holes.