It is fair to say that the field of nonlinear chemical dynamics would not be where it is today, and perhaps it would not exist at all, without fast digital computers. As we saw in Chapter 1, the numerical simulation of the essential behavior of the BZ reaction (Edelson et al., 1975) did much both to support the FKN mechanism and to make credible the idea that chemical oscillators could be understood without invoking any new principles of chemical kinetics. In 1975, solving those differential equations challenged the most advanced machines of the day, yet the computers used then were less powerful than many of today’s home computers! Despite the present widespread availability of computing power, there remain many challenging computational problems in nonlinear dynamics, and even seemingly simple equations can be difficult to solve or maybe even lead to spurious results. In this chapter, we will look at some of the most widely used computational techniques, try to provide a rudimentary understanding of how the methods work (and how they can fail!), and list some of the tools that are available. There are several reasons for utilizing the techniques described in this chapter: 1. For a complicated system, it is generally not possible to measure all of the rate constants in a proposed mechanism. One way to estimate the remaining parameters is to simulate numerically the behavior of the system, varying the unknown rate constants until the model satisfactorily reproduces the experimental behavior. 2. If a mechanism, which may consist of dozens of elementary chemical reactions, is valid, then it should reproduce the observed dynamical behavior. Proposed mechanisms are most commonly tested by integrating the corresponding rate equations numerically and comparing the results with the experimental time series, or by comparing the results of many such simulations with different initial conditions (or of a numerical continuation study) to the experimental phase diagram. 3. Numerical results can act as a guide to further experiments. The real reason for developing models is not to interpolate between our experimental observations but to extrapolate into unknown realms.
Keywords: AUTO package, CONT package, Euler's method, GEAR integration package, Matlab, cellular automata, discretization error, explicit integration methods, global error, implicit integration methods
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