lecture 20

lecture 20 - CE 30125 - Lecture 20 LECTURE 20 SOLUTION TO...

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CE 30125 - Lecture 20 p. 20.1 LECTURE 20 SOLUTION TO SINGLE 1ST ORDER INITIAL VALUE PROBLEMS (IVP’s) •So lv e i.c. • Consider two classes of methods: • Runge-Kutta type formulae • single step methods • very simple to program • self starting (only need i.c.’s) • Multi-step formulae • Multi-step methods are much more efficient than single step methods (for the same accuracy) • Multi-step methods are not self starting use single step method to start up and then go over to multi-step dy dt ----- fyt  = yt o y o =
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CE 30125 - Lecture 20 p. 20.2 Runge-Kutta type formulas Single Step Methods • Solution is obtained in terms of , and evaluated for various values of between and self starting . • Self starting since solution involves only information between therefore all information required is available at the 1st step (i.e. the response function of the previous step only). INSERT FIGURE NO. 91 • Various orders of accuracy are available: • 1st order - Euler • 2nd order - Improved Euler, Modified Euler • 4th order - Runge-Kutta y j 1 + y j fy j t j  fyt y t j t j 1 + t j tt j 1 +  xxx x D t solution known to here solution desired here y y j y j+1 t j+1 t j t
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CE 30125 - Lecture 20 p. 20.3 Multi-step Methods • Require information for in order to predict the value at INSERT FIGURE NO. 92 • Multi-step methods are dependent on several previous conditions • Use F.D.’s in the development of the multi-step formulae • Adams open formula • Adams closed formula • Predictor - Corrector Methods (combination of the above 2 methods) • Multi-step methods are much more efficient than single step methods (for the same accuracy) • Multi-step methods are not self starting use single step method to start up tt j t j 1 + D t y y 0 y j+1 t j+1 t j t t t y j-2 y j-1 y j t j-1 t j-2 t 0
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CE 30125 - Lecture 20 p. 20.4 Runge-Kutta type methods • Solve • Recursive relationship for all Runge-Kutta methods: where ( is therefore the slope as per the definition of ) . . . • We must select ‘s and ‘s • Select these coefficients such that you minimize errors. • Compare Taylor Series expansion of and select the recursive relationship coef- ficients such that you eliminate the appropriate error terms.
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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lecture 20 - CE 30125 - Lecture 20 LECTURE 20 SOLUTION TO...

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