Introduction

Three kinds of mathematical problems have appeared frequently earlier in the book: the time evolution of a pattern-forming system, the identification of stationary states (e.g. a uniform state or a periodic hexagonal lattice), and the calculation of growth rates σq (eigenvalues) for small-amplitude perturbations of a stationary state. Except for simplified mathematical models that often cannot be compared quantitatively with experiment, and except for rather narrow parameter regimes such as just beyond the onset of a supercritical bifurcation, these three classes of problems cannot be solved analytically. It can then be helpful to use numerical methods on a digital computer.

In this chapter, we discuss some numerical ideas and algorithms to solve the first two of these three kinds of problems. The discussion will be useful in several ways. First, many difficult concepts associated with pattern formation such as spatiotemporal chaos can often first be conveniently studied using a numerical method since the alternatives of experiments or analytics can be more time consuming, expensive, or difficult. Second, the great power of current computers and of modern numerical algorithms increasingly allow the investigation of evolution equations that describe a nonequilibrium system quantitatively and sometimes provide the only way to obtain information about a system. Simulations thus complement theory and experiment as an important third way of exploring and understanding nonequilibrium phenomena. Third, the following discussion should help you to understand the assumptions that underlie some of the numerical methods used to study pattern-forming systems and so give you a sense of when you can trust the simulations.