Synopsis for B21a Numerical Solution of Differential Equations I

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 0286)

Recommended Prerequisites:

None.

Overview

To introduce and give an understanding of numerical methods for the solution of ordinary and partial differential equations, their derivation, analysis and applicability.
The MT lectures are devoted to numerical methods for initial value problems, while the HT lectures concentrate on the numerical solution of boundary value problems.

Learning Outcomes

At the end of the course the student will be able to:

1. construct one-step and linear multistep methods for the numerical solution of initial-value problems for ordinary differential equations and systems of such equations, and to analyse their stability and accuracy properties;
2. construct finite difference methods for the numerical solution of initial-boundary-value problems for second-order parabolic partial differential equations, and first-order hyperbolic equations, and to analyse their stability and accuracy properties.

Synopsis

The MT part of the course is devoted to the development and analysis of numerical methods for initial value problems. We begin by considering classical techniques for the numerical solution of ordinary differential equations. The problem of stiffness is discussed in tandem with the associated questions of step-size control and adaptivity:

Initial value problems for ordinary differential equations: Euler, multistep and Runge–Kutta; stiffness; error control and adaptive algorithms.

The remaining lectures focus on the numerical solution of initial value problems for partial differential equations, including parabolic and hyperbolic problems:

Initial value problems for partial differential equations: parabolic equations, hyperbolic equations; explicit and implicit methods; accuracy, stability and convergence, Fourier analysis, CFL condition.

The course will be based on the following textbooks:

1. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge University Press, 1994). ISBN 0-521-42922-6 (Paperback edition) [Chapters 2, 3 (Secs. 3.1, 3.2), Chapter 4 (Secs. 4.1–4.6), Chapter 5].
2. E. Süli and D. Mayers, An Introduction to Numerical Analysis (Cambridge University Press, 2003). ISBN 0-521-00794-1 (Paperback edition) [Chapter 12].
3. A. Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, 1996). ISBN 0-521-55655-4 (Paperback edition) [Chapters 1–5, 13, 14].