The collision of black holes is a classic problem in the strong-field regime of general relativity, with no known closed-form solution. When exact solutions are not known, or perturbative expansions about known solutions are inadequate to capture the non-linear dynamics, the numerical solution of the Einstein field equations is required. The main purpose of this talk is to describe, in detail, the necessary ingredients for achieving stable Cauchy evolution of black hole collisions in asymptotically anti-de Sitter (AdS) spacetimes. I will begin by motivating this program in terms of the heavy-ion physics it is intended to clarify. I will then give an overview of anti-de Sitter space, the mapping to its dual boundary conformal field theory, and the method we use to numerically solve the fully non-linear Einstein field equations with AdS boundary conditions. As a concrete example of these ideas, I will describe simulations of stable AdS black hole mergers in 5 dimensions.