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 + + ) Relatively good accuracy, better convergence ( ) x O x x f x x f dx df o o x + + ) ( ) ( ONE SIDED upwind differencing: The error leads to numerical stability but is a mathematical trick. Adds in error D eff = D true + V x x/2 and Pe local, eff < 2. Still wrong, because artificially increased. V 2 + v V + S( ) = 0 ( ) ) ( ) ( 2 1 2 1 1 = + + + + + x f x v x i i i i i linear linear linear or nonlinear ( ) ( ) ) ( ) ( 2 1 = + % x f x f M ( ) ( ) 2 1 = + % f f M approx. to differential operators Newtons Method F ( ) = 0 J = -F Newton update J = M + % 2 1 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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