Unformatted text preview: r and Step Size Control
We would like to design software that automatically adjusts the step size so that some
measure of the error, ideally the global error, is less than a prescribed tolerance. While
automatic variation of the step size is easy with one-step methods, it is very di cult to
compute global error measures. A priori bounds, such as (3.4.11), tend to be too conservative and, hence, use very small step sizes (cf. 16], Section II.3). Other more accurate
procedures (cf. 15], pp. 13-14) tend to be computationally expensive. Controlling a
42 measure of the local (or local discretization) error, on the other hand, is fairly straight
forward and this is the approach that we shall study in this section.
A pseudo-code segment illustrating the structure of a one-step method yn = yn 1 + h (tn 1 yn 1 h)
; ; (3.5.1a) ; that performs a single integration step of the vector IVP y = f (t y)
0 y(0) = y0 (3.5.1b) is shown in Figure 3.5.1. On input, y contains an approximation of the solution at time
t. On output, t is replaced by t + h and y contains the computed approximate solution
at t + h. The step size must be de ned on input, but may be mod...
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- Spring '14
- Numerical Analysis, yn, Tn, Numerical ordinary differential equations