COSC2500 Numerical Methods in Computational Science 2013 Projects
COSC2500 Numerical Methods in Computational Science 2013
Projects
1 Introduction
The assessment for COSC2500 includes a project. The project is not required to pass the course, and does not contribute towards a grade of 4 or 5. The project will be awarded marks that will contribute towards grades of 6 or 7. The marks will be awarded based on the standard of the project:
standard marks
1–5 0 6 1–10
7
2 Assessment and assessment criteria
11–20
Keep in mind that your project is part of your assessment for the course. Your main task is not to solve the problem you’ve chose as your topic, but rather to show that you have achieved the learning objectives and should be given a good mark. This means that
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A good mark can be obtained even if the problem is not solved.
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A solution will not necessarily demonstrate achievement of more than a very limited range of
learning objectives. A solution that has required a lot of work might show less than a much
quicker and simpler solution that is clever, original, neat, elegant, innovative, etc.
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You can do a lot of stuff in a project beyond simply getting the answer. For example, you can find out what types of methods have been used to answer the problem or similar problems before, compare your computational results with experimental or theoretical results, and so on.
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You need to communicate what you have learned, not just what your result is.
The passing grades, and descriptions thereof are:
Grade 7, High Distinction: As for 6, with consistent evidence of substantial originality and insight in identifying, generating and communicating competing arguments, perspectives or problem solving approaches; critically evaluates problems, their solutions and implications.
Grade 6, Distinction: As for 5, with frequent evidence of originality in defining and analysing is- sues or problems and in creating solutions; uses a level, style and means of communication appropriate to the discipline and the audience.
Grade 5, Credit: Demonstrates substantial understanding of fundamental concepts of the field of study and ability to apply these concepts in a variety of contexts; develops or adapts convincing arguments and provides coherent justification; communicates information and ideas clearly and fluently in terms of the conventions of the discipline.
Grade 4, Pass: Demonstrates adequate understanding and application of the fundamental concepts of the field of study; develops routine arguments or decisions and provides acceptable justi- fication; communicates information and ideas adequately in terms of the conventions of the discipline.
COSC2500 Numerical Methods in Computational Science 2013 2 For the typical project, the following are important:
• Reference sources of information used.
• Appropriate literature review.
• Use the methods in the course to calculate results using a mathematical model.
• Relate your results to the system being modelled.
• Validation and verification.
• Clear communication of results, including visualisation where appropriate.
You could also:
• Compare a number of methods against each other.
• Compare different implementations of the same method.
• Demonstrate an understanding of the mathematics behind the numerical methods you use.
• Demonstrate an understanding of the mathematical behaviour of the model.
Students may, optionally, give a self-assessment of their work with their written report.
• What mark or grade do you think you should be given for this assessment task? Justify this based on the assessment criteria and standards. You may give either a 1–7 grade or a specific mark.
3 Project topics
A typical project would involve the mathematical modelling of some system of interest, the numerical solution of the mathematical model, and discussion of the results (for example, what the results mean, comparison with experimental results, other computational results, and so on). What science are you interested in? What systems have been mathematically modelled in that science, or could be modelled? One approach is to find a paper in the research literature presenting computational results, and attempt to replicate them.
Some possible topics are listed below. These include “typical” projects such as described above. Other types of topics are also listed.
You should write a summary of your topic, with an outline of the work you intend to carry out, and note how this allows you to demonstrate achievement of the learning objectives of the course.
Long-time stability of numerical solution of differential equations
When solving a system of differential equations by stepping forward in time from given initial conditions, a small error per time step can accumulate to become very large. A classic example of a system where this is important is the calculation of planetary orbits about the sun.
Investigate and compare the performance of different methods, such as Euler’s method, higher- order methods such as Runge-Kutta integration, Verlet integration, and so on.
The Death of Economics
The economies of nations are complex, and the modelling of them is a difficult problem. How- ever, the importance of economics has meant that much work and energy has been spent on this problem. Discuss methods that have been attempted, and their success or lack thereof. Give example implementations to show the behaviour of the models involved.
Reference: Paul Ormerod, The Death of Economics, St. Martin’s Press (1994). Polarisability of Platonic solids
Calculations of the dielectric polarisabilities of Platonic solids are presented in A. Sihvola, P. Yla ̈-Oijala, S. Ja ̈rvenpa ̈a ̈, and J. Avelin, “Polarizabilities of Platonic solids”, IEEE Transactions on Antennas and Propagation 52(9), 2226–2233 (2004).
Implement their method, and compare your results with theirs. (The ambitious might like to try this for all of the Platonic solids; a good plan might be to try it for the cube only.)
COSC2500 Numerical Methods in Computational Science 2013 3 Airplane boarding
Simulate the boarding of aircraft by passengers. Test different strategies to minimise the board- ing time.
Finite element software
Obtain the source code of a finite element program. How does it work? Demonstrate the functioning of the code.
(Of course, you could do something like this for other types of software as well, not just a finite-element code.)
The optimal length of a rifle barrel
A model for projectile propulsion in a rifle barrel can be used to predict the muzzle speed for a given barrel length. Use such a model, and compare the results with known experimental results.
Reference: P. Mohazzabi and B. M. Shefchik, “The optimal length of a rifle barrel”, Canadian Journal of Physics 80, 541–440 (2002).
Antenna arrays
A common method of controlling the directionality of radiation of electromagnetic waves by an antenna (described by the radiation pattern) is to use an antenna array, a collection of individual antennas. The total field at any point will be the sum of the fields produced by the individual antennas.
Calculate the field produced by an antenna array. Investigate the design of arrays to produce desired radiation patterns (i.e., the inverse problem).
Methods for integral transforms
A very useful mathematical method is to represent a function f(x) in terms of a basis set of functions ψn(x), such that
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f(x)= ∑anψn(x) (1) n=1
where an are constants (often called expansion coefficients, mode ampitudes, etc.) which spec- ify the function f(x). This is a Hilbert space representation of the function. The best-known examples are Fourier series and Fourier transforms. More generally, this type of representation is also called a generalized Fourier transformation or an orthogonal eigenfunction transformation.
If the basis functions are orthonormal, i.e., we have
Z
ψn(x)ψm(x)dx = δnm, (2) the constants an can be found by integration:
an = Z ψn(x)ψm(x)dx = δnm. (3) This conversion from f(x) as a set of values of the function over x to the constants an is an
integral transform.
Clearly, we can perform the integral transform numerically by integration as per equation (??). However, note that if we choose a sufficiently large set of values of x, equation (??) provides a linear system from which an can be found.
Choose a particular integral transform, and investigate and compare the performance of inte- gration versus solution of linear systems. You might also wish to consider fast transforms, such as the fast Fourier transform (FFT) and similar transformations.
COSC2500 Numerical Methods in Computational Science 2013 4 Numerical methods in cell biology
Cell biology is a diverse field with many possible applications of numerical methods. What types of methods and mathematical models are used? Choose one application and investigate in detail. Implement and test the model if feasible. Is the mathematical model used outside biology?
Computing the eigenvalues of large matrices
Why is it important and why is it difficult? What methods are used?
