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lec13_10112006

# lec13_10112006 - 10.34 Numerical Methods Applied to...

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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 0 ) = 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 error 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 0 )/< t> < 10 8 {FLOOR} Figure 2. If t floor is larger than t maximum , then a solution cannot be found.

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