Peristaltic pumping is a mechanism for transporting fluid or immersed particles in a channel by waves of contraction. It occurs in many biological organisms as well as in several human designed systems. In this study, we investigate numerically the peristaltic pumping of an incompressible viscoelastic fluid using the simple Oldroyd-B model coupled to the Navier-Stokes equations. The pump's walls are assumed to be massless immersed fibers whose prescribed periodic motion and flow interaction is handled with the Immersed Boundary Method. We utilize a new, highly efficient non-stiff version of this method which allows us to explore an unprecedented range of parameter regimes, nearly all possible occlusion ratios and Weissenberg numbers in excess of 100. Our numerical investigation reveals rich, highly concentrated stress structures and new, striking dynamics. The investigation also points to the limitations of the Oldroyd B model, with a potential finite time blow-up, and to the role of numerical regularization. (C) 2012 Elsevier B.V. All rights reserved.