Comparing the Laplace Transform and Parareal Algorithms

Both the Lap lace Transform and the Para real family of algorithms promise to provide completely parallel in time and space computational results. Given a random time dependent partial differential equation it is unclear which algorithm will run faster. We define the algorithms in question, including more specific details than usual. We define interesting parallel environments, some of which do not exist yet. Finally, we demonstrate some computational environments in which one of the algorithms can be expected to be faster than the other algorithm.