lecture15 - Summary from last lecture- Starting from the...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Summary from last lecture- Starting from the Taylors series expansion, we derive:- Eulers method- Improvement of Eulers method- Heuns method- Midpoint method- Higher order Taylor method Higher order Taylor method Euler method is only a first order approximation (use only two terms in Taylors series with truncation error O(h 2 )). It is obvious that if we want a better solution, we should use more terms in the Taylor series in order to obtain a higher order truncation error. Suppose the solution y(t) to the initial-value problem: ) , ( y t f dt dy = for a t b y(t o ) = y o has (n+1) continuous derivatives. If we expand the solution, y(t) in therms of its nth Taylor polynomial about t: 1 ) 1 ( 1 ) ( 2 1 ) ( )! 1 ( ) ( ! ) ( ! 2 ) ( ) ( ) ( + + + + + + + + + = i i i n n i n n i i i i t t y n h t y n h t y h h t y t y t y K Higher order Taylor method 1 ) ( 1 ) 1 ( 2 1 ) , ( )! 1 ( ) , ( ! ) , ( ! 2 ) , ( ) ( ) ( + + + + + + + + = i i i i n n i i n n i i i i i i t t y f n h y t f n h y t f h y t f h t y t y K Successive differentiation of the solution y(t) gives: )) ( , ( ) ( )) ( , ( ) ( )) ( , ( ) ( ) 1 ( ) ( t y t f t y t y t f t y t y t f t y n n = = = M Substitute: ( ) ( ) ( ) + + + = = + i i n n i i i i i i o o w t f n h w t f h w t f h w w y w , ! , 2 , ) 1 ( 1 1 K Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Ordinary differential equations- Runge Kutta Method Derivation of the Runge Kutta scheme Suppose we have a step size x = h and want to estimate y at x+h: o y ) y(x y x f dt dy = = ) , ( Consider the first order differential equation: ( ) ( ) ( ) ( ) 1 1 2 1 1 1 1 2 2 1 2 2 3 1 1 1 2 1 , , , , + + + + = + + + = + + = = n n n n n n k k k y h x f h k k k y h x f h k k y h x f h k y x f h k K M n n k k k y y + + + + = K 2 2 1 1 1 all ks like h . f(x ,y ) where i i i and i are constant which will be adjusted to gives y 1 (or at next step) with an error that goes like h to as high a power as possible....
View Full Document

Page1 / 29

lecture15 - Summary from last lecture- Starting from the...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online