In the Improved Euler method slopes are averaged, but in this method positions of points are averaged as is shown geometrically in Fig. 1.12. We begin with the line L1 that passes through (tm, ym) and has the slope f[tm, ym]. We proceed along L1 and find the point of intersection with the ordinate erected at t„+ h/2; this is the point P at which y = ym+ (h/2) ym'. The slope of the function at this point is then calculated:
The line through P with this new slope is shown as L* in Fig. 1.12. Next we draw a line parallel to L* passing through (tm, ym) that is shown as L0. Now designate the value of ym+1 to be the intersection of L0 with t = tm + h. Hence the equation for L0 is y = ym + (t - tm) f[tm,ym, h] . [1.43]
Equations [1.40 - 1.43] define the modified Euler method that is also known as the improved polygon method, for (almost!) obvious reasons. It can be shown to be a second-order Runge-Kutta method.
Figure 1.12. Graphical analysis of the Modified Euler method of numerical integration.
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