Chemical Engineering Hand Written_Notes_Part_79

Chemical Engineering Hand Written_Notes_Part_79 - 160 4....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
160 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES De f ning (4.27) z ( n )= x ( n 2) x ( n 1) x ( n ) ; z ( n +1)= x ( n 1) x ( n ) x ( n +1) ; B = 010 001 η 2 η 1 η 0 we have z ( n +1) = B z ( n ) (4.28) x ( n z 3 ( n = h i z ( n C z ( n ) (4.29) Similarly, the true solution can be expressed as z ( n B z ( n ) (4.30) x ( n C z ( n ) (4.31) where (4.32) B = e ah 00 0 e ah 0 e ah The evolution of the approximation error is given as e ( n B e ( n )+[ B B ] z ( n ) (4.33) e ( n z ( n ) z ( n ) (4.34) If the stability criterion that can be used to choose integration interval h can be derived as (4.35) ρ ( B ) < 1 Note that characteristic equation for matrix B is given as (4.36) λ 3 η 0 λ 2 η 1 λ η 2 =0 Thus, eigenvales of matrix B can be directly computed using the coe cients η 0 1 and η 2 , which are functions of integration interval h. Equations such as (4.18), (4.22) and (4.36) can be used to generate stability envelopes for each method in the complex plane (eigenvalues of a matrix can be complex). Stability envelopes for most of the methods are available in literature.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_79 - 160 4....

This preview shows document pages 1 - 2. Sign up to view the full document.

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