Modelling: Theory and Practice - 2019 entry

The role of the mathematician has changed significantly with the ubiquity and processing power of modern computers. Drawing on the modelling techniques learned in the first year, you will analyse these more closely, understanding the mathematics that underpins them and the extent to which they are appropriate in different situations. You will also work together to develop and analyse your own computer models, including studying the performance of the underlying algorithms and the limits of their predictive power.

Prerequisite modules: MTH1003 and MTH1002 or NSC1002 (Natural Science Students) or equivalent.
 

Corequisite module: MTH2003.

This module explores the use of computers to solve mathematical problems by means of numerical approximation. The techniques discussed form the basis of the numerical simulation and computer modelling of problems in science and business. The key aim is developing an understanding of the numerical algorithms and we will explore these both theoretically and through case studies that develop further the mathematical modelling techniques learned in MTH1003.

On successful completion of this module, you should be able to:

 

Module Specific Skills and Knowledge:

1 demonstrate a working knowledge of the theory and practical implementation of basic numerical methods;

2 explore applications and ideas underpinning more advanced methods that are developed in third/fourth stage modules and project work;

3 develop and code your own mathematical models with guidance;

4 interpret the outputs from your models, drawing suitable conclusions from your data;

5 evaluate the effectiveness of your models at explaining and predicting the phenomena you are modelling.

Discipline Specific Skills and Knowledge:

6 explore the subject material of the module through diverse applications to areas of science and business;

7 use computation as a natural method for tackling such problems;

Personal and Key Skills

8 demonstrate theoretical and practical mathematical skills, including programming.

9 formulate and solve problems independently;

10 communicate computer results and mathematical derivations effectively.

11 work in teams and use a variety of sources to produce reports and other appropriate scientific outputs.

Root Finding

Bisection, Newton-Raphson and fixed point convergence. Proofs of convergence and non-convergence. Demonstration of convergence and non-convergence using diagrams.

Quadrature and Ordinary Differential Equations (ODEs)

Finite differences, including first and second-order approximations for both the first and second derivative. Timestepping of a first-order ODE using the following methods: forward Euler, leap-frog, Runga-Kutta, Implicit, Adams-Bashforth and Adams-Moulton. Understanding of numerical stability, including identifying the true solution.  Analysis of both accuracy and stability of timestepping methods.

Matrices

The LU decomposition and Gaussian elimination for matrix inversion. Iterative matrix inversion methods: Jacobi, Gauss-Seidel and Successive Over Relaxation. Analysis of convergence of the iterative methods using: (1) method of norms and (2) Spectral radius. The condition number. Calculating eigenvalues using the power method.

Partial Differential Equations (PDEs)

1d diffusion and 1d advection equation as prototypical PDEs. Finite difference methods. Implicitness and possible extensions e.g. semi-lagrangian and montone advection.

Case studies will be developed in a variety of areas of mathematics, science and/or business as part of the coursework, to support and enhance the material formally presented in lectures/tutorials.  These case studies might include, for example: optimisation; Fast Fourier Transform; stochastics; modelling of data.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.


If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

ELE – http://vle.exeter.ac.uk

Reading list for this module: