Hi everyone,

I wanted to share the scoring guide that I will use when grading your project – this is based on the Part 1 and Part 2 project assignments.  I also included my own solutions to the Project Example given in Part 2, if you’d like to compare.

Project Scoring Guide

This project is worth 15 points towards the “OpenLab” portion of your grade, equivalent to three OpenLab assignments.  These points are assigned as follows:

  • (5 points) A working numerical methods calculator using your choice of technology.
    • 3 points – solution correctly implements each method (Euler’s Method, Improved Euler’s Method, Runge-Kutta Method)
    • 1 point – solution displays intermediate points and other important values (k1, k2, etc.)
    • 1 point – ease of implementation (can easily change initial condition, step size, target value)
  • (5 points) Solutions to the Project Example
    • 2 points – value of y(7.1) is approximated by each method
    • 3 points – appropriate step size is chosen for each method
  • (5 points) Writing Assignment
    • 2 points – meet minimum length requirement (300 words)
    • 1 point – Describe your project
    • 1 point – Describe the process
    • 1 point – Why do we need numerical methods? Why is this assignment included in the class?

 

 

Project Example Solutions

The example given in the “Project Part 2” post was:

Project Example.  Given the differential equation dy/dx = \frac{xy}{x-y} and initial condition y(6)=0.8, approximate the value of y(7.1) using Euler’s Method, Improved Euler’s Method, and Runge-Kutta.

For each method, choose a step size that gives four correct digits following the decimal point.  How many steps are required to obtain this level of precision?

NOTE: The actual solution is y(7.1)=3.700936…

In my own solution, implemented first in Google Sheets and later (when I needed more computing power) in Microsoft Excel, I obtained the following results:

Euler’s Method

To obtain four correct digits after the decimal point required:

  • 275,000 steps
  • step size h = 0.000004
  • final approximation y(7.1) = 3.70090152430816

 

Improved Euler’s Method

To obtain four correct digits after the decimal point required:

  • 390 steps
  • step size h = 0.002820513
  • final approximation y(7.1) = 3.70090010561837

 

Runge-Kutta Method

To obtain four correct digits after the decimal point required:

  • 8 steps
  • step size h = 0.1375
  • final approximation y(7.1)=3.70091299278611