lec14_10132006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #14: Implicit Ordinary Differential Equation (ODE) Solvers. Shooting. Implicit ODE Solvers dy / dt = -y y(t=0) = 1 y true = e -t with explicit Euler G = F (Y (t)) for this case, instability if Δ t > 1 with implicit Euler G = F (Y (t+ Δ t)) For Δ t = 2, ) Δ new t*F(Y Y Y old t new t t + = Δ + y new = 1+2(-y new ) 1 1/ 3 1/ 4 x 2 4 x true 3 y new = 1 Î y new = 1 / 3 e -2 =y true y new = 1 / 3 + 2(-y new ) 3 y new = 1 / 3 Î y new = 1 / 9 e -4 =y true Figure 1. Comparison of implicit Euler to true value. Accuracy low, but Implicit Euler does not become numerically unstable. Explicit Euler decays too fast. Implicit Euler decays too slow, but it allows one to use larger timesteps. Stiff Solvers Stiff: t f – t 0 >> Δ t max because of instability because of accuracy Explicit | λ | max Δ t 1 for stability Stiff solvers: ode15s Å usually better ode23s Å super stiff Non-stiff ode45 Å explicit method
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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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lec14_10132006 - 10.34, Numerical Methods Applied to...

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