Simple Oscillating String

Abstract

We have two basic methods of modeling matter. We can treat matter as a continuum and solve differential equations or we can treat it discretely and solve massive N-body problems. The differential equations produced by meaningful problems can be extremely difficult to formulate and much more difficult to solve, if not impossible. Hence, in most cases, the resultant differential equation is discretized and approximated numerically. We know that matter is discrete, so why all the circular work of taking a discrete phenomenon, putting a continuous model on it, and then discretizing this continuous model into something that is solvable? We do this because the sheer number of particles that make up any meaningful amount of matter is daunting and impossible to handle even with today’s largest supercomputer. One discrete approach to deal with this problem is to group large numbers of particles together and treat them as individual units called quasimolecules. Then pray that the bulk behavior of this model behaves in a similar manner to the matter being studied. The mathematical skills needed to set up such problems are much more attainable than those needed to set up the differential equation, but the N-body problem that ensues usually requires a supercomputer to propagate it through time. Until recently the cost of such machines made this approach out of the financial reach of all except the privileged few. But, thanks to the gaming industry and recent advances in the ease to which one can program modern graphics processing units (GPUs), this is no longer true. Supercomputing has finally reached the masses. Here we use a simple example of a vibrating string to compare the continuous and discrete approaches to modeling and show how GPUs can enhance an undergraduate modeling experience.