The simulation of multiphase flow in porous media is a ubiquitous problem in a wide variety of fields, such as fuel cell modeling, oil reservoir simulation, magma dynamics, and tumor modeling. However, it is computationally expensive. This paper presents an interconnected set of algorithms which we show can accelerate computations by more than two orders of magnitude compared to traditional techniques, yet retains the high accuracy necessary for practical applications. Specifically, we base our approach on a new adaptive operator splitting technique driven by an a posteriori criterion to separate the flow from the transport equations, adaptive meshing to reduce the size of the discretized problem, efficient block preconditioned solver techniques for fast solution of the discrete equations, and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. We demonstrate the accuracy and efficiency of our approach using numerical experiments in one, two, and three dimensions using a program that is made available as part of a large open source library.