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. Runge-Kutta
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
; ; 1=2 ; ) + f (tn)] which is the same as Simpson's rule integration.
Our limited experience with Runge-Kutta 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 four-stage, fourth-order Runge-Kutta methods are now clear.
From Table 3.2.1, we see that a fth order Runge-Kutta method requires an additional
two function evaluations per step. Additionally, Butcher 8] showed that an explicit
s-stage Runge-Kutta method will have an order of at least s ; 2.
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- Spring '14
- Numerical Analysis, yn, Tn, Numerical ordinary differential equations