The University of Southampton
Courses

# MATH3018 Numerical Methods

## Module Overview

Introduce the students to the practical application of a relatively wide spectrum of numerical techniques and familiarise the students with numerical coding. Often in mathematics, it is possible to prove the existence of a solution to a given problem, but it is not possible to "find it". For example, there are general theorems to prove the existence and uniqueness of an initial value problem for an ordinary differential equation. However, it is in general impossible to find an analytical expression for the solution. In cases like these numerical methods can provide an answer, albeit limited: for example, there are numerical procedures (called algorithms) that, given an initial value problem, will compute its solution. This module is designed to cover four key areas: linear equations, quadratures (ie the evaluation of definite integrals) and the solution of Ordinary and Partial Differential Equations. The nature of the module is eminently practical: we will cover relatively little of the mathematical background of the numerical techniques that we will study. On the other hand students will be required to do a reasonable amount of programming in a language such as Matlab or Python; part of the assessment will test their ability to code in Matlab or Python and to put into practice the theoretical methods studied at lectures. Computer laboratory sessions are associated to this module and will complement the lectures. One of the pre-requisites for MATH6149

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Analyse a mathematical problem and determine which numerical technique to use to solve it
• Use a programming language such as Python, its instructions and its programming language
• Demonstrate knowledge and understanding of numerical methods to solve systems of linear equations, to compute quadratures and to solve Ordinary and Partial Differential Equations
• Show logical thinking in coding a mathematical problem in algorithmic form
• Use your knowledge of a programming language such as Python to learn more easily any other programming language you will need to use in future

### Syllabus

Linear Systems Linear systems, direct methods (Gaussian and LU decomposition), indirect methods (Jacobi, Gauss-Seidel). Quadratures Polynomial interpolation methods and adaptive methods. Initial Value for Ordinary Differential Equations Basic theory, one-step methods (Euler, Runge-Kutta), predictor-corrector methods, multi-stepmethods (Adam-Bashforth, Adam-Moulton). Higher order ODEs and systems of ODEs. Boundary Value Problems for ODEs Shooting, finite differences. Partial Differential Equations Basic theory, simple PDEs (Poisson, Heat, Wave). Finite difference algorithms for parabolic, hyperbolic and elliptic PDEs. Non-Linear Equations Bisection method. Contraction mappings and Newton’s method for functions of one or more variables. Matlab or Python Introduction, commands to solve quadratures and integrate ordinary and partial differential equations. Basic programming techniques.

### Learning and Teaching

#### Teaching and learning methods

Lectures, computer laboratory, web supported materials and private study.

TypeHours
Independent Study90
Teaching60
Total study time150

### Assessment

#### Summative

MethodPercentage contribution
Class Test 10%
Exam  (120 minutes) 60%
Implementation and testing 30%

#### Referral

MethodPercentage contribution
Exam %

#### Repeat Information

Repeat type: Internal & External

Pre-requisites: MATH2015 OR MATH2038 OR MATH2047 OR MATH2048

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.