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2017/18 Taught Postgraduate Module Catalogue

MATH5476M Advanced Modern Numerical Methods

20 creditsClass Size: 30

Module manager: Dr Evy Kersale
Email: E.Kersale@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2017/18

Pre-requisite qualifications

MATH3474 or equivalent.

Pre-requisites

MATH3474

Numerical Methods

This module is mutually exclusive with

MATH3475

Modern Numerical Methods

This module is approved as an Elective

Module summary

The development of fast, accurate and efficient numerical methods has been a critical area in recent decades. Building upon a basic knowledge of discretisation and linear algebra gleaned in MATH3474, this module aims to provide students with an overview of more sophisticated numerical methods, such as finite elements, collocation and spectral methods, used in the solution of, in particular, elliptic, parabolic and hyperbolic PDEs. As such, the course should provide a sound basis for students considering postgraduate work in diverse areas.

Objectives

On completion of this module, students should be equipped with a sophisticated array of advanced numerical methods that have been developed with efficiency and speed in mind.

Students should be able to implement these methods in order to obtain accurate and economical solutions of ODEs and PDEs arising from the modelling of phenomena in diverse scientific areas. The advanced topics should form a solid foundation for potential research students.

Learning outcomes
To build upon the basic numerical methods introduced in MATH3474 in order to develop more advanced methods to approximate with increasing sophistication the spatial derivatives in PDEs, effectively reducing them to ODEs.

In particular, attention will focus on numerical methods for weak formulations of BVPs, error analyses for strong formulations, and highly accurate spectral methods that have characterised the advances in the subject during recent decades.

Syllabus

Boundary - value problems (14 lectures):
- Strong, weak & variational formulations
- self - adjoint 2nd-order differential operators
- 2-point BVPs; inhomogeneous BCs; basis & trial functions
- defect / residual
- essential & natural BCs
- Galerkin weighted - residual method
- application to general and self - adjoint BVPs
- pintwise residual; nodal collocation
- regular- & Chebyshev - node spectral collocation; variational principles
- Euler - Lagrange equation; energy functionals
- Rayliegh - Ritz method & relationship to Galerkin methed
- inhomogeneous mixed BCs
- Rayleigh - Ritz method for self - adjoint 2 - D BVPs
- finite element basis functions
- linear & quadratic Lagrangian elements
- cubic Hermitian elements
- errors in and convergence of the finite - element method
- Gauss - Legendre quadrature.

Initial - value problems (10 lectures):
- Stability of single - and multi - step methods for ODEs
- linearisation
- reduced equation
- parasitic solutions
- strong, weak, relative & absolute stability
- stability polynomial; complex reduced equation
- hk stability diagram
- parabolic & hyperbolic PDEs
- stability of 1 - D advection equation
- truncation error
- consistency
- equivalent PDE
- rounding error
- convergence
- Lax' equivalence theorem
- eigenvalue stability analysis
- normal matrices
- Fourier / von Neumann error decomposition
- error resolution
- aliasing
- amplification factor
- CFL condition
- dissipation & dispersion
- decay of high frequencies
- spectral error analysis
- dissipation & dispersion errors
- leading & lagging errors.

Spectral Methods (5 lectures):
- Trigonometric interpolation; - discrete Fourier transform
- periodicity
- aliasing explained
- spectral convergence
- spectral differentiation
- interpolation on infinite non - periodic domains
- Whittaker's cardinal function
- trigonometric cardinal functions
- spectral differentiation matrices
- supergeometric convergence
- non-periodic cardinal functions
- spectral Chebyshev interpolation on [-1, 1]
- spectral Chebyshev differentiation matrix.

Integral-Equation Methods (6 lectures, level-5):
- Volterra and Fredholm equations of the first, second and third kinds;
- characteristic values; quadrature rules for marching 1-D VIE1s and VIE2s;
- trial-function approximations for FIE2s;
- exact and approximate solutions;
- iterative Neumann-series method for 1-D FIE2s;
- converged Neumann series;
- direct methods;
- Gauss-Legendre quadrature for FIE2s;
- Nyström method for FIE2s;
- error analysis.

Students may be asked to undertake background reading on an additional advanced topic.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	44	1.00	44.00
Private study hours			156.00
Total Contact hours			44.00
Total hours (100hr per 10 credits)			200.00

Private study

- Consolidation of course notes and background reading;
- Directed reading choice of advanced extra topics, examined by a compulsory examination question.

Opportunities for Formative Feedback

Regular compulsory examples sheets and optional Maple worksheets.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	3 hr 00 mins	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 26/04/2017

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