CO906 Numerical Simulation of Continuous Systems (2008-9)
Course information for the current year is here.
Module Leader: Dr Colm Connaughton (Mathematics and Complexity)
link to Online Course Materials
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:
-
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.
-
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
-
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
-
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
-
Spectral Methods
- Fast Fourier Transform
- Spectral and Pseudo-spectral methods
- Applications: Nonlinear Schrodinger Equation
- Fast Fourier Transform
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 |