Flow visualization techniques involving extreme advection workloads are becoming increasingly popular. While these techniques often produce insightful images, the execution times to carry out the corresponding computations are lengthy. With this work, we introduce an alternative to traditional advection. Our approach centers around block exterior flow maps (BEFMs). BEFMs can be used to accelerate flow computations by reducing redundant calculations, at the cost of decreased accuracy. Our algorithm uses Lagrangian interpolation, but falls back to Eulerian advection whenever regions of high error are encountered. In our study, we demonstrate that the BEFM-based approach can lead to significant savings in time, with limited loss in accuracy.