# 2022/23 Undergraduate Module Catalogue

## MATH3476 Numerical Methods

### 15 creditsClass Size: 100

Module manager: Adrian Barker & Kuan Li
Email: a.j.barker@leeds.ac.uk / earkli@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2022/23

### Pre-requisites

 COMP2421 Numerical Computation MATH2600 Numerical Analysis MATH2601 Numerical Analysis with Computation

Module replaces

MATH3474

This module is not approved as a discovery module

### Module summary

Ordinary and partial differential equations (ODEs and PDEs) are ubiquitous in the modelling of real problems arising in science, engineering and economics. However, only rarely can ODEs and PDEs be solved exactly in mathematical terms, and so approximate methods of solution are of paramount importance.The basic idea employed in this course is that of discretizing the original continuous problem to obtain a discrete problem, or system of equations, that may be solved with the aid of a computer.This course introduces the basic ideas underlying approximation and its application, via finite differences, to the solution of ODEs and PDEs. As part of the approximation process, numerical linear algebraic techniques are developed in order to provide calculable solutions to the discrete equations.

### Objectives

Learning outcomes
On completion of this module, students should be able to:
- work independently to acquire an understanding of the relevant background theory and to apply theory to solve problems involving PDEs in practice;
- interpolate periodic and non-periodic data on a finite 1-D interval using minimax and Chebyshev approximation techniques; understand the Runge phenomenon; understand spectral accuracy;
- approximate partial derivatives by finite differences in both 1-and 2-D to pre-specified orders and accuracy
- set up and solve linear systems of simultaneous algebraic equations for 1- and 2-D elliptic Boundary Value Problems obtained using finite difference methods; compute numerically eigenvalues and eigenvectors of matrices.
- understand time-stepping methods so that they can solve PDE Initial Value Problems, relevant for the most important hyperbolic and parabolic PDEs arising in real-world problems.

### Syllabus

The following topics will be covered:
- Introduction to the canonical types of PDEs and examples of where they arise in real-world problems in science, engineering and economics
- Approximation Theory (Lagrange interpolation; interpolation errors; Chebyshev polynomials and interpolation)
- Numerical Differentiation (1-D finite differences of arbitrary order and accuracy; 2-D FD formulae)
- Elliptic PDEs and numerical linear algebra (sparse systems of linear equations and computational work; solution of elliptic Dirichlet and Neumann BVPs; eigenvalues and eigenvectors of matrices)
- Parabolic & hyperbolic PDEs and time-stepping methods for initial-value problems (1-D advection equation; errors; Fourier / von Neumann error decomposition; CFL condition)

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Private study hours 128.00 Total Contact hours 22.00 Total hours (100hr per 10 credits) 150.00

### Opportunities for Formative Feedback

Interaction with module manager through regular practical classes. Assessment of success on example sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 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