Synopsis for B21b Numerical Solution of Differential Equations II

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 0287)

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

Students will understand and have experience of the theory for: Construction of shooting methods for boundary value problems in one independent variable

Elementary numerical analysis of elliptic partial differential equations analysis of iterative methods for solution of large linear systems of equations

Synopsis

The HT part of the course is concerned with numerical methods for boundary value problems. We begin by developing numerical techniques for the approximation of boundary value problems for second-order ordinary differential equations. Boundary value problems for ordinary differential equations: shooting and finite difference methods. [Introduction (1 lecture) + 2 lectures]

Then we consider finite difference schemes for elliptic boundary value problems. This is followed by an introduction to the theory of direct and iterative algorithms for the solution of large systems of linear algebraic equations which arise from the discretisation of elliptic boundary value problems.

Boundary value problems for PDEs: finite difference discretisation; Poisson equation. Associated methods of sparse numerical algebra: sparse Gaussian elimination, iterative methods. [13 lectures]

This course does not follow any particular textbook, but the following essentially cover the material:

1. A. Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, 1996), Chapters 7,10,11.
2. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge University Press, 1994.
   Or the more recent 2nd edition, 2005), Chapters 6 and 7.
3. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods (Clarendon Press, Oxford, 1985 (and any later editions)), has some of the material in chapter 5.