DATA-log

Current studies have led me to meddle with the famous FPU (Fermi-Pasta-Ulam) problem. Unsuprising event, since it is one of the cornerstones of the study of computational physics.

It was one of the first problems that was tackled not using analytic math tools, but using high speed digital computing. The reason behind this sort of approach was the difficulty of dealing with nonlinear equations; something that is near impossible to deal with exact analytical attacks. Digital computers and numerical analysis however is the ideal tool to conduct these sort of chaotic computational experiments with.

A lot has been written about the FPU problem (try the wikipedia article for a decent summary), but an immediate way to grasp the problem is by hearing how it sounds. The system described in the problem consists of masses coupled together, the usual scalar wave equation with nonlinear coupling terms added. Here the initial gaussian pulse oscillates in the system without damping and with increasing nonlinearity.

Fpu1 by janne808

Another example is done with a custom VST plugin. The system is driven with two pulse oscillators.

Fpu2 by janne808

Connection machines were a line of parallel supercomputers built by Thinking Machines Corporation. Notably Stephen Wolfram and Richard Feynman were involved in the early years of the corporation.

Here are some delightfully academic and stiff Thinking Machines Corp. promo videos from Youtube. The first one features some great footage from a lattice gas automata fluid dynamics model (I can’t believe it took a person year for the LGA model, I put one together in a week or so on MATLAB. My ego is pleased.)

Ironically Thinking Machines Corp. went bankcrupt in 1994, when parallel computing is a hot commodity today in 2009.

Having studied the fundamentals of quantum mechanics from the perspective of numerical computation it has become obvious that stability is a major drawback in most fast numerical approaches. The best scheme to tackle with the issue is by Visscher whose explicit algorithm is both fast and stable. Accuracy is not my biggest concern since I’m mostly interested in visualization of the quantum mechanical world but the algorithm holds very well. The algorithm is second order accurate, there is some dispersion of the wave packet however. It is also straightforward to extend it to n-dimensional space.

My processing example applet of the 1D time-dependent Schrödinger wave equation with the source code is at http://www.punainen.org/~biotek/visscher1d

Reference to the original paper: P.B. Visscher “A fast explicit algorithm for the time-dependent Schrödinger equation”, Computers in Physics Nov/Dec 1991, 596-8

After a few hours of playing around with SSE intrinsics I managed to cut the cpu cycles in half (more or less).

Here’s the link:
http://www.punainen.org/~biotek/string_sse.dll