A simple approach is as follows. For the predictor we use a simple second-order method, namely, ym0+1 = ym-1 + 2hf[tm,ym] , [1.60]
where the superscript (0) indicates that this is the initial guess at ym+1, i.e., the predicted value. Immediately note that the method cannot be used to compute y1 since to do so would require the point y-1. Thus, a Runge-Kutta method is used to predict y1. Alternatively, we might have thought that Euler's method could have been used here, thus obviating the need for (tm-1 , ym-1 ), but it turns out that the truncation error in this method is routinely too large. The use of prior information on the function leads to the classification of these methods as multistep ones.
Figure 1.13 gives some geometrical insight into the operation of the predictor. First, we draw the line L parallel to L1 through the point (tm-1, ym-1). This line intersects the ordinate, erected at t = tm+1, at the predicted value of y„°+1. We now improve this prediction by drawing the line L parallel to L1 through the point (tm-1, ym-1). This line intersects the ordinate erected at t = tm+1 at the predicted value of y„°+1. Then we improve this prediction.
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