	Department of Mathematics & Computer Science

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MA2071 Scientific Computing

MA2071(=MC228) Scientific Computing

Credits: 10

Convenor: Dr Paul Houston

Semester: 2

Prerequisites:	essential: MA1151(=MC146), MA1152(=MC147), MA2101(=MC248)	desirable:
Assessment:	Coursework: 50%	One and a half hour exam: 50%

Lectures:	18	Problem Classes:	5
Tutorials:	none	Private Study:	47
Labs:	5	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	75

Explanation of Pre-requisites

Students have learned in MA1152 about the structure of linear systems of algebraic equations and something of methods for solving such problems. Here they will see how these techniques can be applied to solve problems arising in engineering and science applications. The concept of an eigenvalue (also from MA1152) will be used in the analysis of the Jacobi and Gauss-Siedel iterations. Ideas and techniques from MA1151 and MA2101 are used in error analysis for all types of problems.

Course Description

This course introduces some of the methods and ideas used in applied numerical modelling, such as one typically encounters in chemistry, physics and engineering problem-solving. The problem is usually to determine, for given input data, an approximate solution to an equation or system of equations that is accurate to some desired error tolerance. The challenge is to find ways to do this efficiently and to analyze the effects of various types of errors on the quality of the approximation.

The dimension of systems that arise in real-world applications can be quite large, and it is therefore essential to build algorithms whose complexity scales reasonably with the problem size. Sometimes a particular structure is present in the problem formulation which can improve the numerical treatment. For example, it is often possible to solve symmetric matrices more reliably and accurately than nonsymmetric ones of equivalent dimension.

Many of the most important numerical tasks have to do with approximating the solutions of differential equations. For the purposes of illustrating the process, we restrict attention to a few elementary (scalar) ordinary differential equations, and a few popular schemes for their numerical approximation. The convergence of a given method is dependent on two important properties: the magnitude of the perturbations or local error introduced at each stage of the calculation, and stability which relates to the rate of growth of perturbations through the many steps of a long computational process. We are often faced with the task of comparing two methods, in which case the order of accuracy of the method becomes important.

Besides introducing some of the theoretical issues of numerical analysis, this course will give students a taste of modern scientific computing software tools, especially the MATLAB package which facilitates interactive evaluation of numerical methods and graphical visualization of the results of simulation. As part of the course, students will complete a series of computer projects using MATLAB.

Objectives

By the end of this module the student should:

appreciate the interplay among the elements of modern scientific computing, including modelling, simulation, and error analysis;
be familiar with a few of the basic algorithms for solving linear systems, nonlinear equations, and ordinary differential equations;
gain some measure of familiarity with the software package MATLAB and its use in evaluating numerical algorithms.

Transferable Skills

This course will impart a basic understanding of some of the key methods of scientific computing as well as an introduction to software tools such as MATLAB. These skills are important in all areas of modern applied science and engineering and are also widely used in industry.

Syllabus

1. Models and Methods

Mathematical Models
Ordinary Differential Equations
Discretization
Algorithms
Types of Error and norms.

2. Two-point BVPs and linear systems

Simple models, e.g. $u_{xx} = a u + f(x), \;\; u(0) = u_0, \;\; u(L) = u_L$
Finite difference discretization and linear systems
LU and Cholesky factorization
Jacobi and Gauss-Siedel iterations

3. Nonlinear Equations and Iterations

Models and implicit functions, Root-finding and Kepler's equation.
Fixed point iterations
linear vs. superlinear convergence
Newton's method and 2nd order convergence

Reading list

Details of Assessment

The final assessment of this module will consist of 50% coursework and 50% from a one and a half hour examination during the Summer exam period. The 50% coursework contribution will be determined from students' solutions to coursework problem sheets and a series of MATLAB projects. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.