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(5)	Complexity Science MSc (+PhD)	1	option	12
P-G1P8(9)	Complexity Science MSc (+PhD)	1	option	12
P-G3G1	Maths and Stats MSc (+PhD)	1	option	12
P-F3P6(7)	MSc in Complex Systems Science	1 or 2	option	12

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 1 hours
Classwork sessions per week	2 x 1 hours
Module duration	10 weeks
Total contact hours	40
Private study and group working	80

Assessment information 2011:

Week	Assessment	Issued	Deadline	how assessed	%credit
1	Problem sheet #1	11-01-11	31-01-11 12:00	written script	15
4	Problem sheet #2	01-02-11	21-02-11 12:00	written script	20
7	Problem sheet #3	22-02-11	14-03-11 12:00	written script	15
10	Oral Examination	15-03-11		Oral examination	50