Breadcrumb

Numerical Methods for PDEs (MATHM0011)

Academic Year:

Contents of this document:

Administrative Information

  1. Unit number and title: MATHM0011 Numerical Methods for PDEs
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2012
  6. Unit Organiser: Rich Kerswell, FRS
  7. Lecturer: Rich Kerswell
  8. Teaching block: 1
  9. Prerequisites: MATH20700 (Numerical Analysis 2) and MATH20402 (Applied Partial Differential equations 2) or by permission for graduate students who have taken the equivalent elsewhere.

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. 

Teaching Methods

The unit will be delivered through lectures.

Learning Objectives

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

Assessment Methods

The final assessment mark will be based on a 1½-hour written examination (100%).

Award of Credit Points

Credit points for the unit are gained by passing the unit (i.e. getting a final assessment mark of 50 or over).

Texts

[1] ``Finite Difference and Spectral Methods for Ordinary & Partial Differential Equations'' L.N. Trefethen, webbook http://people.maths.ox.ac.uk/trefethen/pdetext.html

[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.

Syllabus

  1. Numerical Differentiation [2 lectures]:  Interpolation and  finite difference formulae.
  2. Finite Difference Methods [7 lectures]: Parabolic, Hyperbolic and Elliptic equations,  accuracy, stability and convergence, dissipation, dispersion, group velocity, Gauss-Seidel & Conjugate Gradient methods.
  3. Fourier Spectral Methods [3 lectures]: Discrete Fourier Transform, Galerkin and Collocation approaches, aliasing errors and accuracy, FFT, stability.
  4. Chebyshev Spectral Methods [3 lectures]: connections with Fourier approach, Tau method, BVPs, time-dependent PDEs, stability.