  The expedient design of precision components in aerospace and other high-tech
industries requires simulations of physical phenomena often described by
partial differential equations (PDEs) without exact solutions. Modern design
problems require simulations with a level of resolution difficult to achieve in
reasonable amounts of time---even in effectively parallelized solvers. Though
the scale of the problem relative to available computing power is the greatest
impediment to accelerating these applications, significant performance gains
can be achieved through careful attention to the details of memory
communication and access. The swept time-space decomposition rule reduces
communication between sub-domains by exhausting the domain of influence before
communicating boundary values. Here we present a GPU implementation of the
swept rule, which modifies the algorithm for improved performance on this
processing architecture by prioritizing use of private (shared) memory,
avoiding interblock communication, and overwriting unnecessary values. It shows
significant improvement in the execution time of finite-difference solvers for
one-dimensional unsteady PDEs, producing speedups of 2--9$\times$ for a range
of problem sizes, respectively, compared with simple GPU versions and
7--300$\times$ compared with parallel CPU versions. However, for a more
sophisticated one-dimensional system of equations discretized with a
second-order finite-volume scheme, the swept rule performs 1.2--1.9$\times$
worse than a standard implementation for all problem sizes.
