His fourth and fth order formula pair is 0 1 4 1 4 3

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Unformatted text preview: 7 3168 ; 355 33 46732 5247 49 176 5103 ; 18656 1 35 384 0 500 1113 125 192 2187 ; 6784 11 84 35 384 0 500 1113 125 192 2187 ; 6784 11 84 0 5179 7571 393 92097 187 1 ^ 57600 0 ; 339200 2100 40 16695 640 Having procedures for estimating local (or local discretization) errors, we need to develop practical methods of using them to control step sizes. This will involve the selection of an appropriate (i) error measure, (ii) error test, and (iii) re nement strategy. As indicated in Figure 3.5.1, we will concentrate on step changing algorithms without changing the order of the method. Techniques that automatically vary the order of the method with the step size are more di cult and are not generally used with Runge-Kutta methods (cf., however, Moore and Flaherty 20]). For vector IVPs (3.5.1b), we will measure the \size" of the solution or error estimate by using a vector norm. Many such metrics are possible. Some that suit our needs are 1. the maximum norm ky(t)k = 1max jyi(t)j im (3.5.6a) 1 2. the L1 or sum norm ky(t)k1 = 3. and the L2 or Euclidean norm ky(t)k2 = m X i=1 "m X 49 i=1 jyi(t)j jyi(t)j2 #1=2 (3.5.6b) : (3.5.6c) The two most common error tests are control of the absolute and relative errors. An absolute error test would specify that the chosen measure of the local error be less than a prescribed tolerance thus, ~ kdnk A where the ~ signi es the local error estimate rather than the actual error. Using a relative error t...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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