Case Studies

Abstract

We present a number of non-trivial, linear and nonlinear examples of initial boundary value and time-periodic parabolic partial differential equations. Each problem is solved with the appropriate variant of the multigrid waveform relaxation method as well as with “the best” standard parabolic solver. The differences in performance are explained and the theoretical results obtained in the previous chapters are illustrated. It is shown that the waveform relaxation methods are competitive on sequential processors, and that they outperform the standard techniques on parallel machines. In particular we illustrate that on a 16-processor vector hypercube waveform relaxation can be faster than any of the standard approaches by a factor of ten up to forty.

Suppose you want to teach the “cat” concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractile claws, a distinctive sonic output, etc.? I bet not. You probably show the kid a lot of different cats, saying “kitty” each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience.

—R.P. Boas, “Can we make mathematics intelligible” (Am. Math. Monthly 10, p. 727, Dec. 1981).

Although their work differs from the experimental research associated with, say, test tubes and noxious chemicals, mathematicians, like chemists and other researchers, often collect piles of data — whether prime numbers or diagrams of knots — before they can begin to extract and abstract the principles that account for their observations.

—I. Peterson, “Searching for new mathematics”, (SIAM Review Vol. 33, No. 1, pp. 37–42, March 1991).