Synopsis

The case studies in scientific computing form an essential part of the MSc course and take place weekly in Hilary term. During term four topics are studied, each over four weeks. Each week there will be a one hour class to describe some aspect of the topic and tasks will be set. Part of the class will also be used to discuss the previous week's tasks. Students will be able to complete all the tasks using Matlab and the numerical techniques learnt in the Michaelmas term core Numerical Analysis courses or described in detail in the classes. The assessment takes the form of a short report for each topic. This should primarily consist of the material from the weekly tasks along with an explanation of the problem and interpretation of the results. Each case study is worth half a unit and all students must complete at least one.

The first project involves the solution of the Volterra model for population growth in a closed system. This model takes the form of an integro-differential equation and we will solve it numerically using techniques described in the first six lectures of the Numerical Solution of Differential Equations course. Under certain conditions the problem is stiff and a uniform mesh is unsuitable, so we will use the Milne Device to generate the time-steps.

The second topic is concerned with heating one container inside another. We will model the transfer of heat both through both containers. The model will consist of a different heat equation valid in each container with suitable continuity equations at their interface. Solution methods will be those learnt for parabolic partial differential equations in the Numerical Solution of Differential Equations course although special extra techniques will be needed if the container is very thin.

The third project concerns the numerical solution of stochastic differential equations. First we will discuss the ideas of Brownian motion and discretized Brownian paths and then we will discuss methods for evaluating stochastic integrals numerically. We will then use these integral methods to motivate the Euler-Maruyama formula for solving autonomous stochastic differential equations. Finally we will look at Milstein's higher order method.

In the fourth project we will look at the problem of how best to shoot free throws in basketball. First we will try to find the best angle to throw at given that we can throw perfectly at a prescribed velocity but we may make errors in release angle. Then we will consider the best trajectory to throw at to allow us to make errors in both release angle and velocity and still score a basket. We will also consider the effects of air resistance.