Unformatted text preview: error we have to subtract the two Taylor's series representations of the solution.
This is very tedious and typically does not yield a useful result. RungeKutta
methods do not yield simple a priori error estimates.
2. Four function evaluations are required per time step.
3. In the (unlikely) case when f is a function of t only, (3.2.4) reduces to yn = yn 1 + h f (tn 1 ) + 4f (tn
6
; ; 1=2 ; ) + f (tn)] which is the same as Simpson's rule integration.
Our limited experience with RungeKutta methods would suggest that the number
of function evaluations increases linearly with the order of the method. Unfortunately,
Butcher 8] showed that this is not the case. Some key results are summarized in Table
3.2.1. The popularity of the fourstage, fourthorder RungeKutta methods are now clear.
From Table 3.2.1, we see that a fth order RungeKutta method requires an additional
two function evaluations per step. Additionally, Butcher 8] showed that an explicit
sstage RungeKutta method will have an order of at least s ; 2.
Alth...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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