lec20_10272006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #20: Boundary Value Problems Lecture 3. Finite Differences, Method of Lines, and Finite Elements. Finite Differences 0 0 ) ( ˆ D ˆ for difference finite mesh = + ⎯→ = + S A S D φ ϕϕ F i ( φ ) = ( Σ A ij φ j ) + S( φ i ) = 0 i = 1, N mesh *N scalar fields J in = n i F Solve F ( φ ) = 0 by Newton-type methods Need Jacobian J : J in = A in + δ in S’( φ i ). + = % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 0 0 0 0 0 0 J Iterative , need good initial guess φ guess ! (normally, J Δ φ = -F ) Method of Lines e.g. 2D, v y = 0, v x independent of φ : () ) ( ) ( ) ( 0 ) , ( ) , ( 2 2 2 2 , , y x y x x y x x v y v x v y x S y x D x v x y x v i i ϕ + = = + + + + convection diffusion coefficient D replace 0 if v y = 0 2 1 1 ) ( ) , ( ) , ( 2 ) , ( y y x y x y x i i i Δ + + Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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u i (x) = φ (x,y i ) Δ Δ Δ =
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lec20_10272006 - 10.34, Numerical Methods Applied to...

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