Description

This continuously assessed module is delivered as two threads running in parallel. The first develops students' skills in scientific computer programming. The second explores how mathematical descriptions of physical systems can be evaluated and investigated numerically.

The lectures will use language and examples that assume a working knowledge of C (e.g. as provided by PHY2027 Scientific Programming in C) and Octave (e.g. as provided by PHY1028 IT and Electronics Skills).

Module Aims

Computational physics is a subdiscipline lying between expeimental and theoretical physics. Scientists use its techniques to investigate systems that are inaccessible to experiment and/or intractable using the standard methods of theoretical techniques. Students taking this module will develop both their programming skills and their knowledge of a range of computer algorithms of relevance to the simulation and modelling of physical systems.

Other fields have adopted the methodologies discussed in this module. Many computer games, for example, use 'physics engines' make their virtual world behave in a realistic manner. The finance industry employs computational physicists to model the financial markets and the global economy using analagous techniques.

Intended Learning Outcomes (ILOs)

A student who has passed this module should be able to:

• Module Specific Skills and Knowledge:
  1. construct and evaluate mathematical and numerical models of common physical systems and processes;
  2. write and document well-structured scientific computer code of good quality;
  3. write scientific code that can import and export data in a range of formats, e.g. suitable for visualisation by third-party applications;
  4. use mathematics to obtain a computational advantage in the evaluation of solutions;
  5. critically evaluate the success or otherwise of numerical approaches to solving problems;
  6. use profiling tools to identify performance bottle-necks;
• Discipline Specific Skills and Knowledge:
  7. use computational methods to model physical systems;
  8. identify problems that are amenable to computer solution and investigation;
• Personal and Key Transferable / Employment Skills and Knowledge:
  9. undertake co-operative learning by discussing the contents of the module amongst themselves;
  10. write clear and concise descriptions of complicated processes;
  11. balance workload and work independently in order to meet deadlines.

Syllabus Plan

Thread 1: Programming for Physicists

1. The Programmer's Toolkit
   1. Languages: Imperative, functional, constraint; high and low level
   2. Divide and conquer: using pipes and scripting as glue
   3. Useful software libraries for science
2. Simple Data Types
   1. Characters
   2. Integers: unsigned and signed
   3. Floating Point Numbers
   4. Pointers: names and values
3. Functions, Procedures and Objects
   1. Variables and scope
4. Memory
   1. The stack; the heap; allocators; reference counting; garbage collection
   2. Hierarchy: registers, processor cache, random access memory (RAM), disk
5. Data structures
   1. Arrays, character strings, linked lists, queues, stacks, trees, hash tables
6. Input and output
   1. Command Line Interface (CLI) vs Graphical User Interface (GUI)
   2. Exchange formats for data: plain text, binary, FITS, XML
   3. Graphics and visualisation: Gnuplot, POV-Ray, etc.
7. Algorithms
   1. Classification, Big-O notation
   2. Discrete Fourier transform (DFT), fast Fourier transform (FFT)
   3. Quicksort vs Mergesort
   4. Fused multiply add (FMA), Kahan summation
8. High performance code
   1. Code profiling and manual optimisation
   2. Optimisations performed by compilers
   3. Code examples that run (a) quickly, (b) slowly
9. Parallel Programming
10. Program Design and Maintenance
    1. Anticipated usage and user community
    2. Version control, documentation, release strategy, ethics, licences
    3. Code quality: assert statements, compiler warnings, static analysis, crash reports, bug tracking, test driven development, code reviews
    4. Single processor vs multi-processor

Thread 2: Numerical Methods with Physical Applications

1. Numerical Errors
   1. Error Propagation: rounding errors; forwards error analysis; backwards error analysis;
   2. The IEEE-754 standard: precision, exceptions, rounding
2. Interpolation, Differentiation and Integration
   Lagrange polylominals, Runge's phenomenon; Gaussian quadrature, Newton-Cotes forumulae, Richardson extrapolation; Romberg's method
3. Solution of Nonlinear Equations
   Bisection methods, Secant Method, Newton's method, Brent's method, coincident roots and deflation
4. Matrix Algebra
   Simple matrix problems; sparse matrices; systems of equations and matrix inversion; error propation; direct vs indirect methods; matrix eigenvalue problems
5. Ordinary Differential Equations
   1. Reduction of order-N equation to a set of N order-1 coupled ODEs
   2. Methods: forward Euler, backwards Euler, trapezoidal, higher-order methods
   3. Consistency, zero-stability, convergence, A-stability
   4. Step-size control, stiff equations
   5. Classical 1-, 2-, 3-, and N-body problems
6. Boundary Value Problems
   1. Types of boundary condition: Dirichlet, Neumann, mixed
   2. Methods: shooting, multiple shooting, finite difference, finite element
   3. Poisson's equation
7. Partial Differential Equations
   
   1. The diffusion equation
   2. The wave equation
   3. Particle in cell methods
8. Monte Carlo Methods and Simulation
   1. Random number generation
      1. Speed and quality of pseudo random number generators
      2. Uniform distribution
      3. Arbitrary distributions: inversion method, acceptance-rejection method
      4. Gaussian distributions: Box-Muller method
      5. Sub-random sequences
   2. Error estimates
   3. Variance reduction
   4. Percolation
   5. Magnetic systems
9. Function Minimisation and Maximisation
   1. Levenberg-Marquardt method
   2. Nelder-Mead (Simplex) method
   3. Constraints
   4. Genetic algorithms
   5. Simulated annealing
   6. Curve fitting
10. Case Study - Spice 3f5

Advanced Topics (If Time Permits)

1. The Quantum One-Body Problem
2. The Quantum N-Body Problem
   1. Quantum Chemistry Methods
   2. Exact Methods
3. Molecular Dynamics
4. Electronic Structure
   1. Variational methods
   2. Hartree-Fock theory
   3. Density functional theory
5. Polymers and Neurons
6. Quantum Monte Carlo Methods
   1. Systems of Fermions
   2. Bose-Einstein Condensation
   3. Diffusion Monte Carlo
7. Quantum Field Theory

Learning and Teaching

Learning Activities and Teaching Methods

Assessment

Resources

The following list is offered as an indication of the type & level of information that students are expected to consult. Further guidance will be provided by the Module Instructor(s).

Core text:

• Kernighan B.W. and Richie D.M. (1988), The C Programming Language (2^nd edition), Prentice Hall, ISBN 0-13-110362-8

Supplementary texts:

• Atkinson K.E. (1989), An Introduction to Numerical Analysis (2^nd edition), John Wiley, ISBN 978-0471500230
• DeVries P.L. and Hasbun J.E. (2010), A First Course in Computational Physics, Jones and Bartlett
• Giordano N.J. and Nakanishi H. (2006), Computational Physics (2^nd edition), Pearson, ISBN 978-0-131-46990-7
• Press W.H., Teukolsky S.A., Wetterling W.T. and Flannery B.P. (2007), Numerical Recipes (3^rd edition), Cambridge University Press, ISBN 978-0-521-88068-8
• Thijssen J. (2007), Computational Physics (2^nd edition), Cambridge, ISBN 978-0521833462

ELE:

• http://newton.ex.ac.uk/redirects/ELE/phym004

Further Information

Prior Knowledge Requirements

Re-assessment

Re-assessment is not available except when required by referral or deferral.

Notes: See Physics Assessment Conventions.

KIS Data Summary

Miscellaneous