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Chemical Engineering Hand Written_Notes_Part_79

Chemical Engineering Hand Written_Notes_Part_79 - 160 4...

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160 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES De fi 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 = 0 1 0 0 0 1 η 2 η 1 η 0 we have z ( n + 1) = B z ( n ) (4.28) x ( n + 1) = z 3 ( n + 1) = h 0 0 1 i z ( n ) = C z ( n ) (4.29) Similarly, the true solution can be expressed as z ( n + 1) = B z ( n ) (4.30) x ( n + 1) = C z ( n ) (4.31) where (4.32) B = e ah 0 0 0 e ah 0 0 0 e ah The evolution of the approximation error is given as e ( n + 1) = 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 , η 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.
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