The choice of the initial value of h in the predictor corrector method can be guided by the inequality h < 2/M. However, this is not very convenient because the value of M must be estimated, so almost invariably h is simply chosen to be a small fraction around 0.01% of the maximum time of the simulation. Once the numerical integration has started, the automatic procedures specified above 'kicks in.'
If the value of the expression on the left of Eqn [1.63] does not satisfy the inequality, then h is halved and the predictor step is repeated. This is then followed by the corrector step and its iteration. If the value is smaller than required, h is increased usually by doubling it.
Finally, we must consider the manner in which the truncation error grows; in other words, we must consider the instability of the solution. This aspect is addressed by the proper choice of step-size h; and in various Mathematica functions the integration algorithm performs this task automatically, so you will not be unduly troubled by this problem.
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