Milne Predictor Corrector

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

There are a number of important predictor-corrector equations, including equations (3.3.15) and (3.3.18), which are the fourth-order Milne predictor-corrector formulas. 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

M71 Solution of ordinary differential equations predictor-corrector method of Milne 7100 7188... 

An important question is the relative numerical efficiency of the two methods or, more generally, the two families of methods. At a fixed step size the predictor - corrector methods clearly require fewer function evaluations. This does not necessarily means, however, that the predictor - corrector methods are superior in every application. In fact, in our present example increasing the step size leaves the FTunge - Kutta solution almost unchanged, whereas the Milne solution is deteriorating as shown in Table 5.1. 

This is Milne s predictor-corrector method. To ensure greater accuracy, we must first improve the aecuracy of the starting values and then subdivide the intervals. 

Values of y , y , y 2 and y 3 are required to caleulate y , . Milne s method uses Newton-Cotes formula for the predictor and Simpson s rule for the corrector. 

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