Continuous Valued Cellular Automata for Nonlinear Wave Equations

In 1955, E. Fermi, J. Pasta, and S. Ulam investigated quadratic and cubic schemes for numerically simulating nonlinear waves. In our paper, we derive the partial differential equations corresponding to the Fermi-Pasta-Ulam schemes, present a discussion of the accuracy and stability of different schemes for the equations, and implement the schemes as continuous-valued celluar automata. Some illustrative runs of the shareware CAPOW cellular automata simulator are presented which demonstrate the behavior of the different schemes both in stable and unstable domains. These runs include a confirmation of an observation of Fermi, Pasta and Ulam regarding the approximate temporal periodicity of the quadratic and cubic nonlinear wave equations.