Adams-Bashford multistep equation

The advantage of these techniques is that they require only one (additional) function evaluation for each step compared with four or five evaluations for the typical Rimge-Kutta method. Thus, they should be faster. Their principle disadvantage is that they are not self-starting. Another method, Euler s or Runge-Kutta, is required to calculate the first (three) values at x, (x + h), and (x + 2h) before the Adams-Bashford multistep equation can be used to continue the calculation. Additional starting values are also required whenever the step size is changed. 

An extension of the multistep methods is the predictor-corrector approach. Here, the Adams-Bashford equation may be used to calculate a predicted value for y at (x + h). Then a second, corrector, equation is used to refine the valueof y. If the difference between the predictor and corrector values is within specified error limits, thecalculation is continued to the next step, otherwise the step size will be adjusted to maintain the error limits specified. With fewer function evaluations per step, these methods can be faster than the Runge-Kutta methods however, they are not self-starting. 

Multistep Methods. Another class of methods, called multistep methods, involves the use of more than one previous value in the calculation of the next value. With the slope-point and Runge-Kutta methods, a single starting point (and the differential equation) is all that is required for the calculation of the next value at (x + h). With a commonly used four-step method, values at (x — 3h), (x — 2h), (x — h), and x are required in the calculation of y at (x + h). One such method is that of Adams and Bashford. ... 

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