# Notes on Lab #5

## Comparing Numerical Simulation Methods

For this lab we will return to building our own simple models, and
we will compare three sorts of results: (1) analytical computation (correct
value); (2) simulation in Vensim using Euler's Method; (3) simulation in Vensim
using Runge-Kutta 4. We will do this for two different models. (The lab
is a variant on Excerices 6.2.1,3 and 6.4.1,3.)

### Model 1

First, launch Vensim and use it to model
the system *dP/dt* = 0.10*P* with *P*_{0} = 100.
Have t going from 0 through 20, using Euler's method, with Δt = 2.

Then launch Excel, and use it to compute the analytical solution to the
equation for the corresponding times (0, 2, ..., 18, 20).

Returning to Vensim, use the Control Panel to create a vertical table of the population
over time, as you did in a previous lab.
Next, use the copy-to-clipboard button in the table window
to copy the table into Excel, making sure that the rows line up properly.
Then use Excel to compute the relative error at each time.
Then do a plot in Excel of the relative error over time for
Euler. Now repeat the process with RK4 as your integration type.
For the RK4 plot, do one plot with the Y axis limits
as they first appear; then do another plot with the limits set to the same as the
Euler-method plot, so you can see the dramatic difference between Euler and RK4.

Finally, Create a new spreadsheet (or new section of your current spreadsheet)
showing the same four-way comparison (analytical, Euler error, RK4 error, RK4 error with same axes
as Euler) for Δt = 0.25. Looking at the RK4 error plot on its own Y axes (zero
to a very tiny amount), do you see the same orderly increase as with the Euler plot?
If not, can you speculate about what might be going on?
### Model 2

Now repeat this procedure for new model:
*dP/dt* = 0.5(1-P/1000)*P* with *P*_{0} = 20.
Do you remember what sort of model this is? The analytical solution is
on page 209 (Exercise 3).
Comment briefly on what you see, including the difference between the
two models.
### Turnin

For this lab, you only have to turn in the PDF writeup with the plots and
commentary, not the model or Excel files.