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 in a semester

4H: Numerical Methods (MATHS4109)

Assessment

20% course work and 80% exam.

Reassessment

Resit opportunities available for MSc students.

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) Use results from real analysis and linear algebra to study iterative methods for the solution of nonlinear equations (including systems of nonlinear equations);

(b) Derive and use Lagrange interpolation, Newton's iterative method and divided differences to construct polynomial interpolants to given data;

(c) State, prove and apply the error formula and its form when using Chebyshev economisation;

(d) Perform Gaussian Elimination to find the LU decomposition of a matrix;

(e) Analyse iterative methods for systems of linear equations and prove results on convergence.

(f) Construct and analyse Newton-Cotes and Gaussian quadrature schemes to approximate integrals;

(g) Derive finite difference approximations to derivatives and use them to solve BVPs and IVPs for ODEs;

(h) Use finite difference approximations to disctretise the 2D Laplacian and 2D bi-Laplacian.

(i) Construct approximations to solutions of elliptic PDEs with a variety of boundary conditions and in a variety of domains by the finite difference methods;

(j) Derive and analyse the stability of various time-stepping schemes;

(k) Discuss the shooting method.

(l) Construct finite difference approximations to solutions of parabolic PDEs in 1D and 2D;

(m) Analyse the stability and consistency of the numerical methods presented;

(n) Apply the numerical methods presented in the course to produce approximate solutions to stated problems;

(o) Use appropriate software to implement and critique some of the numerical methods studied;

Summary of additions/enhancements, as compared to the level 4 "Numerical Methods":

ILO (a) includes methods for systems of nonlinear equations (not covered in great detail in 4H Numerical Methods).

ILO (j) includes the stability of methods for IVPs

ILO (k) additional discussion of the shooting method.

ILO (l) includes 2D parabolic methods (ADI)

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