The University of Southampton
Courses

# MATH6141 Numerical Methods

## Module Overview

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 eg python; part of the assessment will test their ability to code in a suitable language and to put into practice the theoretical methods studied at lectures. Seven computer laboratory sessions are associated to this module and will complement the lectures.

### Aims and Objectives

#### Module Aims

To introduce the students to the practical application of a relatively wide spectrum of numerical techniques and familiarise the students with their implementation, using eg python.

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Demonstrate knowledge and understanding of numerical methods to solve systems of linear equations, to compute quadratures and to solve Ordinary and Partial Differential Equations
##### Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Use a programming language such as Python, its instructions and its programming language
• 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.
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Analyse a mathematical problem and determine which numerical technique to use to solve it
• Show logical thinking in coding a mathematical problem in algorithmic form

### 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 Problems 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. Introduction to finite elements 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. Programming Use of a suitable language and environment (eg python). Implementation of standard methods. Use of standard libraries.

### Learning and Teaching

#### Teaching and learning methods

36 lectures supported by computer labs, online material and independent study.

TypeHours
Independent Study104
Teaching46
Total study time150

### Assessment

#### Summative

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

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

### 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:

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.

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