Unconstrained Growth

This learning module accompanies chapter two of Angela B. and George W Shiflet's book Introduction to Computational Science. Here, we introduce and develop our ability to:

• Define and use functions in Python.
• Use Jupyter Notebooks to develop and document problem solving process and solution.
• Use Matplotlib to generate good plots.
• Analyze simulation error.

The assignment below will help you as you develop these abilities. Try and complete the assignment on your own or with guided help from a classmate and/or instructor. If you need help, you may reference the notebook Unconstrained Growth Solution which contains a solution to the assignment. Also, the module UnconstrainedGrowthSolution.py has code to model unconstrained exponential growth using finite difference techniques. Only use these documents if you are completely stuck. You will learn best if you complete the assignment without looking at the solutions. These documents are provided so you can consult them if absolutly necessary.

Assignment

Investigate solving dP/dt=rP using numerical methods

• Write a python function that uses finite difference methods to simulate the population growth. Like the example in your text, use P_0=100, r=0.1 and a time step size of 0.005, Δt=0.005. You may want to write out pseudo code before you write in python. Your function should:

  • Take at least the time step size, dt, as a passed parameter.

  • Use a loop structure to create a list of time and population. Note: I used a while loop. Also, I found it easier to plot the values if I stored the time values in one list and the population values in another list.

  • Return the lists from the function. Note: I called my function grow. The last line of my function is return t_array, population_array. I then call the function and get the results with t, p = grow(0.005).

• Using the Juppyter notebook Unconstrained Growth.ipynb which has some starter code:

  • plot your simulation results for population vs. time. Add an appropriate title and axis labels to this graph.

  • In your previous graph, have the population data displayed with red hexagon markers with a transparent face. You might want to check the matplotlib documentation for plot. Also, setting the color alpha value to zero will make the marker transparent as described on this stackoverflow discussion.

  • On a single graph, have plots for the population vs time for different time step sizes, Δt. Create at least four different plots with Δt values ranging from 0.005 to 1.

  • Create a plot of the simulation error vs time for the simulations you created in the previous step. Note that the simulation error is the difference between the exact value and the simulation value. For this, you may want to look at the range function in python.