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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:

  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