lecture 22

lecture 22 - CE 30125 - Lecture 22 LECTURE 22 MULTI STEP...

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CE 30125 - Lecture 22 p. 22.1 LECTURE 22 MULTI STEP METHODS • Solve the i.v.p. • Multi step methods use information from several previous or known time levels INSERT FIGURE NO. 100 dy dt ----- fyt  = yt o y o = y t y 0 t 0 t 1 t 2 t 3 t 4 y 1 y 2 y 3 y 4
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CE 30125 - Lecture 22 p. 22.2 • Open Formulae (Adams-Bashforth) • explicit (non-iterative) • can have stability problems • Closed Formulae (Adams-Moulton) • implicit (iterative) • much better stability properties than open formulae • Predictor-Corrector Methods • 1 cycle predictor open formula • 2-3 cycles corrector closed formula • superior to either open or closed formulae separately
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CE 30125 - Lecture 22 p. 22.3 Open Formulae Derivation • Develop a forward Taylor series of about • However by definition and etc., thus (1) • Now replace the various derivatives of with backward difference approximations 1st Order Accurate Adams Open Formula • Retain only the first two terms in Equation (1) • Same as the “explicit” or 1 st order Euler method y t j y j 1 + y j t + y · j t  2 2! ------------ y ·· j t 3 3! y ··· j +++ = y · j f j = y j f · j = y j 1 + y j tf j t 2! ----- f · j t 2 3! ------------ f j ++ +   + = f j y j 1 + y j t fy j t j + =
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CE 30125 - Lecture 22 p. 22.4 2nd Order Accurate Adams Open Formula • Use a backward difference approximation for • Substituting we obtain: f · j f · j f j f j 1 t ------------------ t 2 ----- f ·· j O t  2 ++ = y j 1 + y j tf j t 2 ----- f j f j 1 t t 2 f j O t 2 t 2 3! ------------ f j    + = y j 1 + y j t 3 2 -- f j 1 2 f j 1 5 12 t 3 f j O t 4 + = y j 1 + y j t 3 2 f j 1 2 f j 1 O t 3 = y j 1 + y j t 3 2 fy j t j 1 2 j 1 t j 1 O t 3 + + =
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CE 30125 - Lecture 22 p. 22.5 INSERT FIGURE NO. 101 •No t e s • Method is second order since the local truncation term is (recall the effect of cumulative error during time stepping) • Formula was derived by developing a forward Taylor series for about and using a backward finite difference approximation for the first derivative of • Note that the method is explicit i.e. the new time level value is computed using the slope at the current and previous time levels and • This formula is not self starting ! Use 2nd order Runge-Kutta (R.K.) method to start the computations y(t) t t j-1 t j t j+1 y j-1 y j y O t  3 y j 1 + y j fy j t j j 1 + j j 1
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CE 30125 - Lecture 22 p. 22.6 Example Application of 2nd Order Adams Open Formula • Problem • i.c. gives us , • Apply 2nd order R.K. (Improved Euler) to start the calculations Now we know , dy dt ----- fyt  = y t o y o = y o t o t 1 t o t + = y 1 * y o t f y o t o + = y 1 y o t 1 2 -- f y o t o fy 1 * t 1 +  + = y 1 t 1
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CE 30125 - Lecture 22 p. 22.7 • From time level to ; apply 2nd order Adams Open Formula Now we know , • From time level to ; apply 2nd order Adams Open Formula Now we know , j 1 = j 1 +2 = t 2 t 1 t + = y 2 y 1 t 3 2 -- f y 1 t 1  1 2 f y o t o + = y 2 t 2 j 2 = j 1 +3 = t 3 t 2 t + = y 3 y 2 t 3 2 f y 2 t 2 1 2
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lecture 22 - CE 30125 - Lecture 22 LECTURE 22 MULTI STEP...

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