5M: Advanced Numerical Methods MATHS5042

  • Academic Session: 2018-19
  • School: School of Mathematics and Statistics
  • Credits: 20
  • Level: Level 5 (SCQF level 11)
  • Typically Offered: Either Semester 1 or Semester 2
  • Available to Visiting Students: No
  • Available to Erasmus Students: No

Short Description

The course will introduce students to a variety of advanced numerical methods including, but not restricted to, numerical linear algebra and the numerical solution of PDEs via finite element, finite difference and spectral methods. Algorithms for problems in optimisation (such as graph theory, linear and nonlinear programming, computational geometry) will be introduced.

Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

5 hours of computer laboratory work

Requirements of Entry

Mandatory Entry Requirements

 

L4 Numerical Methods

 

Recommended Entry Requirements

Assessment

Assessment

60% Final Exam

40% Coursework

 

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.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? No

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. 

Course Aims

The course aim is to introduce students to a variety of advanced numerical methods. In the section on Numerical Linear Algebra topics such as

 

(a) Matrix factorisations

(b) Least squares problems

(c) Eigenvalue problems

(d) Linear systems

 

will be covered. In the section on Numerical Methods for PDEs, topics such as

 

(a) Finite element methods (weak formulation, choice of elements and

basis functions, error estimates, examples in 1D & 2D)

(b) Difference methods for parabolic and hyperbolic PDEs (stability and

the CFL condition, upwinding, finite volume methods for hyperbolic

problems)

(c) Spectral methods

 

will be covered. Other ideas in optimization and computational geometry will also be covered.

Intended Learning Outcomes of Course

On completion of this course students should be able to

 

1. State the properties of a variety of matrix factorisations and be able to

compute the factorisations for small examples.

 

2. State and derive methods for the approximation of certain eigenvalues for

A  GLn(R).

 

3. Formulate methods for solving least squares problems.

 

4. State the weak form of certain elliptic PDEs and show how to derive the

Finite Element Method.

 

5. Identify and analyse numerical methods for parabolic equations.

 

6. State finite difference and flux conservative schemes for the solution of hyperbolic PDEs and analyse their stability.

 

7. Use appropriate computational software to investigate numerical methods.

Minimum Requirement for Award of Credits

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