R

i-i where different choices of the parameters c, and b„ (subject to the constraints of Eq. (17-9c) define different methods. The increment Junction, <!>, is a weighted average of R evaluations of f(t,y) at different points in the integration interval. Note that an R-stage method involves R function evaluations. The constants are always chosen to give the maximum order (and thus minimum truncation error) for a given R; this order is R for R= 1,2,3,4; R-l for R = 5,6,7; and < R-2 for R > 8 [Butcher, 1965]. For this reason, fourth-order four-stage Runge-Kutta methods are the most popular. It requires much tedious algebra to derive the relations among the parameters of a four-stage method that make it of order 4. This derivation leaves two free parameters, resulting in a twofold infinity of fourth-order methods^Ralston, 1965]. One popular choice is*

•This method reduces to Simpson's rule if / is independent of y.

This is the algorithm implemented in subroutine RUNGE in Section 20.3. The chief drawback of Runge-Kutta methods is the many function evaluations required per integration step.

We now turn to a discussion of multistep integration methods. A k-step multistep method has the form k k - 1 ^„+1 = ^2 Pjfn+ i+j-k~ 2 + !+,•-* (17-11)

y-o y=o where different choices of the parameters a- and fy define alternative methods. Depending on the choice of these parameters, a A:-step method requires up to k back values of /„ and y„. One drawback of these methods is that they are not self-starting; some other method, often Runge-Kutta, must be used to calculate the first k values of y„ and /„. Another disadvantage is that step size changes are more difficult than for single-step methods; additional back values must be available if the step size is increased, and intermediate back values must be calculated by interpolation if the step size is decreased.* A third penalty is increased computer storage requirements. The chief advantage of multistep methods is that only one function evaluation is needed per integration step.

A multistep method is explicit if /?* = 0 and implicit if (ik # 0. Implicit methods may appear to be of dubious value, because they apparently require a knowledge of f„+i=f(tn+i,y„+i) to evaluateyn+i. If the original differential equation is linear, however,

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