1. Assignment guidance

In this coursework, you will use a combination of your implementations and software libraries to explore different numerical schemes and you will analyse the methods and results.

2. Assessment tasks

In this coursework, you will be writing a report for your boss at BigNumComp Inc helping them choose a numerical method. You should assume she has the knowledge of a second year undergraduate Computer Science student at the University of Leeds.

Your boss wants to find effective numerical methods for working our how fast a firework should be fired so that it explodes at a certain height.

First, the firework is modelled using a differential equation in terms of its height y(t) and velocity v(t), as shown in Fig. 1:

Here a is the unknown initial upward velocity of the firework. We denote by y(t; a) the solution of the differential equation with initial velocity a.

The firework explodes after a fixed time T. The problem is to determine the value a which would mean the firework explodes at a height H. Equivalently, we can say this mathematically as finding the value a which solves the nonlinear equation

F(a) = y(T; a) − H = 0.

Your boss, helpfully, suggests two test cases for you to consider:

Task

You recognise this problem as being solvable using some methods you have seen in the course and know there are other methods available too. In your preliminary research you find a company internal solver for systems of differential equations (attached as solvers.py). However, your company does not currently have anything useful for nonlinear equations.

You will solve both parts of the task using three methods from the solvers.py library and implement the bisection and secant methods for part (b). You will analyse the results using techniques from the module and make a recommendation about which combination gives the best results and how to best choose parameters.

You should include all the sections in the template in your report. There is guidance of what to include in each section and a guidance word limit for each section too. Code, tables and equations do not count in the word limit. The word limit is only guidance and no penalties will be introduced for going over the limit. You should aim to write less than the word count to ensure your writing is concise and understandable to your audience.

You should submit a jupyter notebook including all computations as your report.

You should write in full sentences throughout to guide the reader through what you are doing. There is no need to include the file solvers.py in your solution. You should write text in Markdown blocks and include all code for the implementation and generating results in Code blocks. You should submit your evaluated jupyter notebook. Your final submission should be less than 1MB.

Library documentation

• "Heun"

• "Ralston"

• "Van der Houwen"

• "SSPRK3"

• "Runge-Kutta"

• "3/8-rule"

• "Ralston-4".

Sample code

Download the file solver.py and place it in the same folder as your code.

An example for solving the a single differential equation:

Solution template

(a) Implementation. First, write code to be able to solve the differential equationsusing the three methods you have chosen for arbitrary initial conditions (y(t =0), v(t = 0)), time step and stopping time (dt, T) and model parameters (m, k, g).

Then, write code to solve the nonlinear problem for arbitrary H and T, your three choices of differential equation solver and all the other differential equation solverparameters (y(t = 0), dt, m, k, g). [100 words]

3. General guidance and study support

Examples of how to approach each aspect of this coursework is given in lectures and using the online notes.

Further support for this assessment is given through the MS Class Team. Details of further support sessions will be given closer to the deadline.

4. Assessment criteria and marking process

Your work will be assessed on your code implementation, your results and their pre sentation, your analysis of the method and results, and your writing quality. Work will be marked as a final assessment for this module so your mark will only be given back as part of your final grade.

The quality of written English will be assessed in this work - further details in the Rubric below. As a minimum, you must ensure:

These are pass/ fail criteria. So irrespective of marks awarded elsewhere, if you do not meet these criteria you will fail overall.

6. Submission requirements

7. Academic misconduct and plagiarism

• Leeds students are part of an academic community that shares ideas and develops new ones.
• You need to learn how to work with others, how to interpret and present other people’s ideas, and how to produce your own independent academic work. It is essential that you can distinguish between other people’s work and your own, and correctly acknowledge other people’s work.
• All students new to the University are expected to complete an online Academic Integrity tutorial and test, and all Leeds students should ensure that they are aware of the principles of Academic integrity.
• When you submit work for assessment it is expected that it will meet the Univer sity’s academic integrity standards.
• If you do not understand what these standards are, or how they apply to your work, then please ask the module teaching staff for further guidance.

By submitting this assignment you are confirming that the work is a true expression of your own work and ideas and that you have given credit to others where their work has contributed to yours.

Presentation of results (15 marks)

Writing (5 marks)

A External library reference