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 5.2.1,3 and 5.4.1,3.)

Model 1

First, open up Vensim and use it to model the system dP/dt = 0.10P with P0 = 100. Have t going from 0 through 20, using Euler's method, with Δt = 2.

Then open up 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 graph of population, and display it as a table. You'll want to use Modify/Running Down to get it into columns, and you'll want to use scientific notation to keep Vensim from representing millions with the letter M. 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, expressed as a percentage. Then do a chart (plot, graph) in Excel of the relative error. Repeat the Vensim part using Runge-Kutta 4 instead of Euler's method, and compute the relative error for RK4 next to the relative error for Euler.

Finally, Create a new spreadsheet showing the same three-way comparison ( analytical, Euler, RK4) for Δt = 0.25.

Model 2

Now repeat this procedure for new model: dP/dt = 0.5(1-P/1000)P with P0 = 20. Do you remember what sort of model this is? The analytical solution is on page 167 (Exercise 3). Comment briefly on what you see, including the difference between the two models. If you're feeling ambitious, go back to the first model and plot the relative error for the final condition (RK4, &Deltat = 0.25) on its own scale, and comment on what you think is going on.

Turnin

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