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2006/07 Undergraduate Module Catalogue

MATH3473 Numerical Solutions of Partial Differential Equations

15 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2006/07

Pre-requisites

(MATH2360 and MATH2200), or MATH2430 or MATH2431 or equivalent

This module is approved as an Elective

Module summary

Informal description: Partial differential equations (PDEs) are ubiquitous in the modelling of real problems arising in science, engineering and economics. However, only rarely can PDEs be solved exactly in mathematical terms, and so approximate methods of solution are of paramount importance. The module will cover two distinct techniques for solving a wide range of PDEs arising "naturally" in the above broad disciplines. Both techniques invoke the basic idea of discretizing the original continuous problem to obtain a discrete problem, or system of equations, which may be solved with the aid of a computer. They are Finite-Difference Methods and Spectral Methods.See the schools website or contact: a.slomson@leeds.ac.uk for more information.

Objectives

To introduce students to the techniques and methodologies for solving differential equations numerically and to teach students how to critically assess the accuracy and viability of various finite difference representations of same partial differential equation. On completion of this module, students should be able to: a) write down finite difference equations which are consistent with the partial differential equations; b) test the finite-difference equations for stability and convergence; c) solve, in principle, large sets of linear equations using a variety of direct and indirect methods; d) deal with singularities in elliptic equations; e) know how to choose an optimum relaxation parameter; f) deal with non-linear partial differential equations.

Syllabus

Of the several methods for solving PDEs numerically, the most direct employs finite difference schemes. This module describes such schemes, emphasising why care must be taken to ensure convergence and stability. The schemes apply to all types of PDE, including many which are non-linear. The approach uses analytic methods to establish useful criteria. The topics covered are: 1. Finite-difference formulae. Descriptive treatment of elliptic equations; Descriptive treatment of parabolic and hyperbolic equations; Finite-difference approximations and notation. 2. Parabolic equations: finite-difference methods, convergence and stability. Explicit method; Crank-Nicolson implicit method; Gauss's elimination method; weighted average approximation; truncation error; consistency; convergence; stability; co-ordinate systems; Richardson's deferred approach to the limit; tridiagonal matrices; non-linear equations. 3. Hyperbolic equations and characteristics. Finite-difference methods on a rectangular mesh; Lax-Wendroff method; Courant-Friedrichs-Lewy condition; characteristics. 4. Elliptic equations and systematic iterative methods. Finite-difference equations in various co-ordinate systems; derivative boundary conditions near a curved boundary; higher-order difference schemes; iterative techniques; relaxation parameters; singularities, non-linear equations.

Teaching methods

Lectures: 26 hours. 7 examples classes.

Methods of assessment

One 3 hour examination at end of semester (100%).

Reading list

The reading list is available from the Library website

Last updated: 26/04/2007

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