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lec19_10252006

# lec19_10252006 - 10.34 Numerical Methods Applied to...

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Unformatted text preview: 10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #19: Boundary Value Problems (BVPs) Lecture 2. Finite Differences ( 2 ) ( 2 ) ( ) ( x O x x x f x x f dx df o o x Δ + Δ Δ − − Δ + ≡ ) Relatively good accuracy, better convergence ( ) x O x x f x x f dx df o o x Δ + Δ − Δ + ≡ ) ( ) ( ONE SIDED upwind differencing: The error leads to numerical stability but is a mathematical trick. Adds in error D eff = D true + V x Δ x/2 and Pe local, eff < 2. Still wrong, because artificially increased. V 2 φ + v · V φ + S( φ ) = 0 ( ) ) ( ) ( 2 1 2 1 1 = + Δ − + Δ + − + − + x f x v x i i i i i φ φ φ φ φ φ linear linear linear or nonlinear ( ) ( ) ) ( ) ( 2 1 = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + × % x f x f M φ φ φ ( ) ( ) 2 1 = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + × % φ φ φ f f M approx. to differential operators Newton’s Method F ( φ ) = 0 J Δ φ = -F Í Newton update J = M + ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ % 2 1 φ φ φ φ f f Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Cite as: William Green, Jr....
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lec19_10252006 - 10.34 Numerical Methods Applied to...

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