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2.3.1 The Adams-Bashforth and Adams-Moulton Formulas     continued...

The Adams-Moulton formula of order p is more accurate than the Adams-Bashforth formula of the same order, so that it can use a larger step size; the Adams-Moulton formula is also more stable. A modern code based on such methods is more complex than a Runge-Kutta code because it must cope with the difficulties of starting the integration and changing the step size. With enough ``memorized'' values, however, we can use whatever order formula we wish in the step from . Modern Adams codes attempt to select the most efficient formula at each step as well as to choose an optimal step size h to achieve a user-specified accuracy.

Figure 7: t-points at which order p Adams-Bashforth and Adams-Moulton formulas interpolate f.

Some general rules-of-thumb about how to choose between Runge-Kutta methods and Adams methods for solving nonstiff problems are given below:

Recent developments in Runge-Kutta methods have shifted these boundaries somewhat; RKSUITE, for example, has an interpolation capability that makes it more efficient than the previous generation of Runge-Kutta codes and the (7,8) solution pair is very efficient at stringent error tolerances.

An excellent discussion of Adams methods as well as a widely used suite of codes is given by Shampine and Gordon in [4]. Gear's test [5] presents a variety of methods, and is a primary source about the solution of stiff problems to be discussed in the next section.