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
with explicit Euler
ytrue = e-t
G = F(Y(t))
for this case, instability if Δt > 1
G = F(Y
with implicit Euler
(t+Δt))
old
Yt +new
+ Δt*F(Y new )
Δt = Yt
ynew = 1+2(-ynew)
3ynew = 1 Î ynew = 1/3
1
1/3
For Δt = 2,
e-2=ytrue
ynew = 1/3 + 2(-ynew)
x
3ynew = 1/3 Î ynew = 1/9
x
1/4
true
2
e-4=ytrue
4
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: tf – t0 >> Δtmax
because of instability
because of accuracy
Explicit |λ|maxΔt ≤ 1 for stability
Stiff solvers:
ode15s Å usually better
ode23s Å super stiff
Non-stiff
ode45 Å explicit method
Example:
CO + ½O2 Æ CO2
In the presence of H2, H2O
OH + CO Æ H + CO2
H + O2 Æ OH + O
1/λOH ~ 10-9 s
-9
Δtexplicit ≤ 10
1/λCO ~ 1 s
s
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical
Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
9 orders of magnitude difference in time scales
In diffusion problems λfast/λslow ~ Nmesh2 ~ 1/(Δx)2 so a fine mesh makes the problem very
stiff.
Shooting
Figure 2.
|-------------------| x
y = height of cannonball
⎛ Y0known ⎞
Y(t=0)= ⎜ guess ⎟
⎟
⎜Y
⎠
⎝ 0
Yi(tf) = Yspecial
Y(tf) Å ode15s(…,Yos,…)
Root-finding (Newton, Broyden)
g(Yguess) = 0
g = Yi(tf) – Yspecial
Yi(tf) Å ode15s(…, Yguess,…)
Ybestguess = bisect(@g, [Ylowguess,Yhighguess],tol)
|g(Ybestguess)|<ftol
|Ybestguess – Ytrueguess| < xtol
function error = g(Yguess)
Y0 = […, Yguess]
Yf = ode15s(@F,Yo,tf,tol,options)
error = Yf(nspecial) - Yspecial
inside ode’s
events Î stop integrating when something happens
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 14
Page 2 of 2
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical
Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].