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Unit information: Numerical Methods for Partial Differential Equations in 2015/16

Please note: you are viewing unit and programme information for a past academic year. Please see the current academic year for up to date information.

Unit name Numerical Methods for Partial Differential Equations
Unit code MATHM0011
Credit points 10
Level of study M/7
Teaching block(s) Teaching Block 1A (weeks 1 - 6)
Unit director Professor. Kerswell
Open unit status Not open

MATH20700 (Numerical Analysis 2) and MATH20402 (Applied Partial Differential equations 2) or by permission for graduate students who have taken the equivalent elsewhere.



School/department School of Mathematics
Faculty Faculty of Science


Unit aims

The aims of this unit are to provide an introduction to a variety of numerical methods for solving partial differential equations. The emphasis will be on understanding the fundamentals: the appropriateness of a given method for a given type of PDE (elliptic, parabolic, hyperbolic) and how to construct an accurate and stable numerical scheme to produce answers of the required precision.

General Description of the Unit

Partial differential equations (PDEs) are ubiquitous in modelling physical systems but are not generally solvable in closed form. This unit will discuss some of the numerical methods used to approximate the solutions of some generic PDEs. Topics will include finite difference methods (spatial discretisation, accuracy, stability and convergence, dissipation and dispersion), spectral methods (approximation theory, Fourier series and periodic problems, Chebyshev polynomial and non-periodic problems, Galerkin, collocation and Tau techniques)

NOTE: This unit is also part of the Oxford-led Taught Course Centre (TCC), and is taken by first- and second-year PhD students in Bristol and its TCC partner departments. The unit has been designed primarily with a postgraduate audience in mind. Undergraduate students should not normally take more than one TCC unit per semester.

Further information is available on the School of Mathematics website:

Intended learning outcomes

A student successfully completing this unit will be able to:

  • explain and apply discretisation methods for PDEs using finite differences (both temporally and spatially);
  • demonstrate familiarity with the phenomena of advection, dissipation and dispersion;
  • appreciate the concepts of accuracy, stability and convergence,
  • know von Neumann stability analysis and be familiar with the Lax Equivalence Theorem;
  • demonstrate familiarity with Fourier spectral methods through the discussion of relevant illustrative examples and the correct selection and use of appropriate analytic techniques;
  • demonstrate familiarity with Chebyshev spectral methods through the discussion of relevant illustrative examples and the correct selection and use of appropriate analytic techniques.

Transferable Skills

Computational techniques; interpretation of computational results; relation of numerical results to mathematical theory.

Teaching details

The unit will be delivered through lectures.

Assessment Details

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

  1. ``Finite Difference and Spectral Methods for Ordinary & Partial Differential Equations`` L.N. Trefethen, webbook
  2. ``A Practical Guide to Pseudospectral Methods`` B. Fornberg, CUP 1998.
  3. ``Chebyshev and Fourier Spectral Methods``, J.P. Boyd, Dover 2001.
  4. ``Spectral Methods in Matlab``, L.N. Trefethen, SIAM 2000.
  5. ``Computational Partial Differential Equations using Matlab``, J. Li & Y.-T. Chen, CRC Press 2009.
  6. ``Numerical Solution of Partial Differential Equations``, K.W. Morton & D.F. Mayers, Cambridge 2005.
  7. ``Numerical Linear Algebra``, L.N. Trefethen & D. Bau, SIAM 1997.