Gas accretion onto black holes is thought to power some of the most energetic astrophysical phenomena observed. Black hole accretion disks are efficient engines for converting binding energy into light, and for launching relativistic unbound flows (jets) such as in gamma ray bursts, microquasars and radio-loud active galactic nuclei (AGN). Some systems individually exhibit a wide variety of spectral and bolometric states while others remain remarkably predictable. As
their brightest emission usually emanates near the black hole's event horizon, they serve as excellent environments for exploring different theories of gravity or for constraining the black hole's geometry. In this talk I will explain how investigators use modern general relativistic magnetohydrodynamic computer simulations to understand accretion observations and probe the strong-field regime of gravity.
In particular, I will focus on three topics. First, I will describe how dynamical models of the accretion flow around Sagittarius A*, the supermassive black hole at the center of our galaxy, can help us predict what we will see when observations at the sub-horizon scale are made soon for the first time. Second, I will explain recent
developments in simulating cooled thin disks and how their results may affect estimates of black hole spin from the disks' thermal spectra. Last, I will describe how temporal variability analysis of our dynamical simulations can offer insight into the common behavior seen in high-energy emission from black holes with masses of 10 solar masses to a billion solar masses.