Lec13_10112006 - 10.34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture#13 Stiffness MATLAB® Ordinary

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Unformatted text preview: 10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #13: Stiffness. MATLAB® Ordinary Differential Equation (ODE) Solvers. From Last Lecture: Numerical Integration dY /dt = F (Y ) Y (t ) = Y 0 G : estimated time average slope from t Æ t+ ∆ t General Algorithm: Y (t+ ∆ t) = Y (t) + ∆ t*G (Y ) G = (time avg. slope) + δ e r r o r Rectangle Rule: Explicit Euler G = F (Y (t)) EXPLICIT Trapezoid Rule: G = ½(F (Y (t)) + F (Y (t+ ∆ t)) IMPLICIT unknown δ ~ O(( ∆ t) m ) want ∆ t È Requirement for accuracy sets ceiling on ∆ t Figure 1. Linear approximation to a function. MATLAB ode45 Runge-Kutta: G formula where error scales ( ∆ t) 5 If ∆ t is small, error is small, but takes many steps (tradeoff) * new t Å t+ ∆ t Adding big numbers and small numbers Æ lose log 10 (N timesteps ) sig figs as ∆ t decreases. This can be a significant problem. If computer has 14 sig figs If you want 6 sig figs in Y (t f ): N timesteps < 10 8 (t f – t )/< ∆ t> < 10 8 {FLOOR} Figure 2. If ∆ t floor is larger than ∆ t maximum , then a solution cannot be found. , then a solution cannot be found....
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.

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Lec13_10112006 - 10.34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture#13 Stiffness MATLAB® Ordinary

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