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CR3 (Submission phase)
Submission phase
Instructions for submission
Using your preferred code editor (e.g. VSCode), in a Python script called "CR3.py", write code to answer the following problem.
Problem
Consider the following ODE, describing the displacement of an oscillator (e.g. a mass attached to a spring) at time away from its
resting position:
where is the angular frequency of the oscillator, given as a fixed parameter. The initial conditions are given as
, the initial displacement,
, the initial velocity.
We seek to solve this equation numerically, using a finite difference method. We discretise time with a step size , and denote by the
approximation to computed at time step , that is .
We approximate the second derivative in the ODE with , to obtain the difference equation:
Rearranging this equation allows us to compute using values computed at the two previous time steps, and . We can initialise
and by discretising the initial conditions:
,
Then, we can compute , followed by , etc., for as many time steps as we desire.
Note that this method provides a "valid" solution as long as . This is called the stability condition.
Your task
Write a function "oscillator(w0, u0, v0, nmax, dt)" which takes 5 input arguments:
a positive number "w0", representing the parameter ,
two numbers "u0" and "v0", representing the initial conditions and respectively,
a positive integer "nmax" greater than or equal to 2, representing the total number of time steps,
a positive number "dt", representing the step size ,
and returns a Numpy vector "U" with a total of "nmax" elements, where the th element is the value of , the approximated solution for the
oscillator displacement at the th time step, computed using the method described above.
Dashboard / My courses / www.learn.ed__81855_1 / Code reviews / CR3
Computing and Numerics (2020-2021)[SEM2]
Setup phase Submission phase
Current phase 
Submit your work
Open for submissions from
Monday, 8 March 2021,
2:00 PM (5 days ago)
Submissions deadline:
Monday, 15 March 2021,
12:00 PM (2 days left)
Assessment phase
Open for assessment from
Monday, 15 March 2021,
4:00 PM (2 days left)
Assessment deadline:
Monday, 22 March 2021,
12:00 PM (9 days left)
Grading evaluation phase Closed
Additionally, before computing the result, the function should check the value of and display a message if necessary:
if , display a message warning the user that the stability condition has been violated. Your message should include the maximum
value of "dt" allowed for the chosen value of "w0", to help your user choose a better step size next time.
if , display a message warning the user that the chosen step size is at the stability limit.
In either case, you should still compute and return the solution.
Testing
After the function definition, write a few tests to check that your function is working. Your first test should use the following values:
You can compare the computed solution to the exact solution, which is given by:
You could plot the exact and computed solutions on the same graph over time -- for valid step sizes (such that ), the two curves should
be close, but they will never be exactly overlapping; the peak values will often be where you see the largest error. For values of and close
to the stability condition (but still valid), you should see that the computed solution oscillates slightly faster than the exact solution. As we have
seen in many other situations, decreasing the step size should generally improve accuracy.
Try different values for the initial conditions and , and check that the first two values of your computed solution are set correctly. Try different
values for the frequency , increasing it should produce a faster-oscillating solution.
Make sure you check that the warning messages appear for appropriate values. Plot the computed solution for both problematic cases -- this is
what you should see:
when (and only when) , the amplitude of the oscillations in the computed solution should grow exponentially over time, and you
should see the first warning message.
when (and only when) , the amplitude of the oscillations in the computed solution should grow linearly over time, and you should
see the second warning message.
when (and only when) , the computed solution should oscillate without growing linearly or exponentially over time, and you should
not see any message.
You should include your test code in the script CR3.py, after the function definition. Make sure you label any plots you produce appropriately
and clearly.
Scope
This is for the more experienced programmers amongst you: remember that this is peer-assessed, and that Computing & Numerics is a course
designed for beginner programmers. Please try to stick to what we've seen in the course material so far, even if you feel confident in doing
something more advanced -- keep it simple.
You should not need to -- but if you absolutely want to use something we haven't seen in the course, then please keep it to a minimum, and make
sure you explain clearly what you are doing (and why) in the code comments, so that a student with little or no previous programming
experience can assess your work with confidence. If you don't follow these guidelines, and submit code which requires much more advanced
knowledge/experience with Python to fully understand than we have seen in the course so far, then your assessors will be able to penalise you
(see marking scheme below).
Submission
Upload your file "CR3.py" here when you are finished. Make sure you upload your file before Monday 15th March, 12pm (noon), otherwise you
won't be able to participate to the peer-assessment phase, and you will get a zero for the whole CR3 task!
Marking scheme
The marking scheme which you will use when assessing next week is given in the table below. Each criterion counts equally towards the grade,
and is assessed on a scale of 0 to 3.

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