# CO906 Numerical Simulation of Continuous Systems (2008-9)

Course information for the current year is here.

Module Leader: Dr Colm Connaughton (Mathematics and Complexity)

Taken by students from:

 Code Degree Title Year of study core or option credits P-F3P4 Complexity Science MSc 1 core 10 P-F3P5 Complexity Science MSc+PhD 1 core 10

Context: This is part of of the Complexity DTC taught programme.

Module Aims:

The module covers computational methods for solving partial differential equations with an emphasis problem solving and applications in Complexity Science.

Syllabus:

1. Basic theory of ordinary differential equations and their numerical solution

• initial and boundary value problems, dynamical systems
• approximation of derivatives via finite differences, error analysis
• Euler and predictor-corrector methods
• Runge-Kutta methods
• Stiffness, instability and singularities
• Applications: Predator-Prey models, chaotic dynamics.
2. Partial differential equations (PDE’s)

• classification of PDE’s as elliptic, parabolic or hyperbolic
• Non-dimensionalisation and similarity solutions
• first order PDE’s and the Method of Characteristics
• Applications: traffic flow models
3. Numerical Solution of Parabolic PDE’s

• Finite difference approximation of the heat equation and explicit Euler method
• Explicit vs implicit time-stepping, stability and the Crank-Nicholson method
• Applications: Black-Scholes equation, Fokker-Planck equation
4. Numerical Solution of Hyperbolic PDE’s

• Explicit methods and CFL criterion
• Implicit methods for second order equations
• Conservation laws and Lax-Wendroff schemes
• Applications: Telegraph Equation
5. Spectral Methods

• Fast Fourier Transform
• Spectral and Pseudo-spectral methods
• Applications: Nonlinear Schrodinger Equation

Illustrative Bibliography:

• J.M. Cooper: Introduction to Partial Differential Equations with MATLAB
• T. Pang: An Introduction to Computational Physics
• W. F. Ames: Numerical Methods for Partial Differential Equations
• Printed lecture notes will also be provided.

Teaching:

 Lectures per week 2 x 2 hours Classwork sessions per week 2 x 2 hours Module duration 5 weeks Total contact hours 40 Private study and group working 60

Assessment information 2009:

 Week Assessment Issued Deadline how assessed %credit 1 Problem sheet #1 08-01-09 19-01-09 10:00 written script 12.5 2 Problem sheet #2 15-01-09 19-01-09 10:00 written script 12.5 3 Problem sheet #3 22-01-09 02-02-09 10:00 written script 12.5 4 Problem sheet #4 29-01-09 02-02-09 10:00 written script 12.5 6 Oral Examination 05-02-09 Oral examination 50