  We explore an approach due to Nievergelt of decomposing a time-evolution
equation along the time dimension and solving it in parallel with as little
communication as possible between the processors. This method computes a map
from initial conditions to final conditions locally on slices of the time
domain, and then patches these operators together into a global solution using
a single communication step. A basic error analysis is given, and some
comparisons are made with other parallel in time methods. Based on the
assumption that parallel computation is cheap but communication is very
expensive, it is shown that this method can be competitive for some problems.
We present numerical simulations on graphics chips and on traditional parallel
clusters using hundreds of processors for a variety of problems to show the
practicality and scalability of the proposed method.
