## MATH3474 Numerical Methods

### 10 creditsClass Size: 60

Module manager: Professor Mark Kelmanson
Email: M.Kelmanson@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### Pre-requisite qualifications

MATH2600 or MATH2601 or COMP2647, or equivalent.

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

On completion of this module, students should be able to:
- interpolate periodic and non-periodic data on a finite 1-D interval using minimax, Chebyshev and forced-oscillation approximation techniques;
- understand the Runge phenomenon; understand spectral accuracy; approximate partial derivatives by differences in both 1-and 2-D to prespecified orders and accuracy using both series and operator methods;
- set up linear systems of simultaneous algebraic equations to solve 1- and 2-D elliptic BVPs;
- solve such equations by a variety of direct and iterative methods; understand the theory underlying such methods.

### Syllabus

Approximation Theory (11 lectures) - Lagrange interpolation; Newton divided differences; interpolation errors; Weierstrass' theorem; minimax approximations; Chebyshev equioscillation and de la Vallee-Poussin theorems; Chebyshev polynomials; least-squares, near-minimax, interpolation; forced-oscillation approximations; spectrally accurate evaluation of Fourier co-efficients.

Numerical Differentiation (5 lectures) - 1-D finite differences of arbitrary order and accuracy; FD operators; implicit FD formulae; regular and irregular meshes; molecules and stencils; 2-D FD formulae; first- and higher-order approximations to Laplacian; Poisson equation and Mehrstellenverfahren; high-order multidimensional derivatives.

Numerical Linear Algebra (6 lectures) - matrix and vector norms; spectral radius; diagonal dominance; Gerschgorin's and Bauer's theorems; sparse systems; tridiagonal systems and Cholesky factorisation; Jacobi, Gauss-siedel and SOR iteration; theoretical convergence estimates; optimum over-relaxation; theoretical optimum for 2-cyclic matrices; solution of elliptic Dirichlet and Neumann BVPs; chessboard enumeration; Richardson extrapolation.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 27 1.00 27.00 Private study hours 73.00 Total Contact hours 27.00 Total hours (100hr per 10 credits) 100.00

### Private study

Consolidation of course notes and background reading.

### Opportunities for Formative Feedback

Weekly personal contact with lecturer in examples classes to discuss/provide assistance with regular question sheets.

Assessment of success on examples sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 1 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