Course Catalogue

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This course introduces students to the analysis and application of numerical methods to solve continuous problems. In particular numerical methods for root-finding, polynomial interpolation, the solution of linear systems, integration and the solution of initial and boundary value problems for ordinary and partial differential equations, will be studied in detail.

34 x 1 hr lectures and 12 x 1 hr tutorials and 5 lab hours in a semester.

Mandatory Entry Requirements

None.

Recommended Entry Requirements

Assessment

20% course work and 80% exam.

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The main aims of the course are to introduce the concept of an approximate solution to a continuous problem, to explore how one might construct such approximations, to qualitatively discuss the errors, stability and rates of convergence of numerical methods, and to investigate some of the methods using appropriate software.

By the end of this course students will be able to:

(a) Derive and explain the methods of interval bisection and Newton-Raphson for the approximation of solutions to one equation in one unknown;
(b) Derive and apply fixed-point iterative methods;
(c) Define the order of convergence of an iterative method;
(d) State the meaning of a function being Lipschitz and use this information to prove results on the convergence properties of fixed-point iterative methods;
(e) Convert divergent iterative schemes to convergent iterative schemes;
(f) Derive and use Lagrange interpolation, Newton's iterative method and divided differences to construct polynomial interpolants to given data;
(g) State, prove and apply the error formula and its form when using Chebyshev economisation;
(h) Perform Gaussian Elimination to find the LU decomposition of a matrix;
(i) Derive iterative methods for the solution of Ax=b;
(j) Define the spectral radius of a matrix and use it to prove convergence of iterative methods;
(k) Construct and analyse Newton-Cotes and Gaussian quadrature schemes to approximate integrals;
(l) Apply single and multi-step explicit methods to solve initial value problems for systems of ODEs;
(m) Derive finite difference approximations to derivatives and use them to solve BVPs for ODEs;
(n) Use finite difference approximations to disctretise the 2D Laplacian;
(o) List the elements of elliptic, parabolic and hyperbolic PDEs (very brief);
(p) Use finite difference approximations to disctretise the 2D Laplacian;
(q) Construct approximations to solutions of elliptic PDEs with a variety of boundary conditions and in a  variety of domains (time-permitting) by the finite difference methods;
(r) Derive various time-stepping schemes;
(s) Construct finite difference approximations to solutions of parabolic PDEs;
(t) Analyse the stability and consistency of the numerical methods used;
(u)Use appropriate software to implement some of the numerical methods studied.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.