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## MATH3475 Modern Numerical Methods

### 15 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-requisites

 MATH3474 Numerical Methods

### This module is mutually exclusive with

This module is not approved as a discovery module

### 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 armed 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 forumulations 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; pointwise residual; nodal collocation; regular- & Chebyshev-node spectral collocation; variational principles; Euler-Largrange equation; energy functionals; Rayleigh-Ritz method & relationship to Galerkin method; 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 (3 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.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Consolidation of course notes and background reading.

### 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) 2 hr 30 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