Dynamical systems are a principal tool in the modeling, prediction, and control of physical phenomena ranging from heat dissipation in complex microelectronic devices, to vibration suppression in large wind turbines, to storm surges before an advancing hurricane. The direct numerical simulation may be the only possibility for accurate prediction or control of such complex phenomena. However, an ever-increasing need for improved accuracy requires the inclusion of more detail at the modeling stage, leading inevitably to larger-scale, more complex dynamical systems. Such systems are often linked to the spatial discretization of underlying time-dependent systems of coupled partial differential equations, and their simulation can create large demands on computational resources.