1 ce 30125 lecture 21 p 2111 insert figure no 99

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CE 30125 - Lecture 21 p. 21.11 INSERT FIGURE NO. 99 • Notes • Find the point by using the weighted average of the 4 slopes • Note that there are other coefficients possible for 4th order Runge-Kutta • We require 4 evaluations of the slope for this 4th order method 4 times the work of the 1st order Runge-Kutta Method y y j+1/2 t j+1 = t j t y j t j + D t 2 t j + D t y j+1/2 * * * y j+1 * * use further improved value of slope to obtain y j +1 3 f(y j + 1/2 , t j + 1/2 ) ** 3 use improved slope to evaluate the new midpoint y j +1/2 ** 2 f(y j + 1/2 , t j + 1/2 ) * 2 f(t j + y j ) used to estimate y j+1/2 * 1 , y j 1 +
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CE 30125 - Lecture 21 p. 21.12 Summary of Runge-Kutta Methods Self starting and interval can be changed at any time without complications Very easy to program Comparable if not better accuracy than other methods Require much more computer time than other methods of comparable accuracy since • Number of functional evaluations is proportional to the accuracy • Functional evaluations can not be reused Local truncation errors are difficult and expensive to obtain (easier for other methods) Δ t
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CE 30125 - Lecture 21 p. 21.13 Qualitative basis for verifying accuracy of solutions use 2 different time steps (similar to Romberg integration) • Can estimate truncation error as: where solution found using (therefore 2 steps) solution found using (1 step) = order of the method • Need to run the solution using two different time steps!
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  • Fall '08
  • Westerink,J
  • dt, Euler, Runge–Kutta methods, t j, INSERT FIGURE NO.

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