Computational Methods for Partial Differential Equations

23-27 January 2006

To introduce the student to a range of numerical methods for the solution of partial differential equations (PDEs).

On completion of this module the student should be able to:

• Understand the properties and behaviour of the different types of PDEs.
• Derive finite difference approximations to linear PDEs of all types, check for consistency, convergence and stability, and solve the resulting system.
• Understand the additional complications in hyperbolic systems.
• Derive finite element or boundary element approximations to an elliptic PDE.
• Assess the relative merits of finite difference, finite element and boundary element methods for the solution of a particular PDE problem.

• Introduction:
  Examples of PDEs. Classification of second order linear PDEs; characteristics; boundary conditions.

• Finite difference methods for elliptic PDEs:
  Discretisation of elliptic operators; truncation error and convergence; iterative methods for the solution of the resulting algebraic system.

• Parabolic PDEs:
  Explicit and implicit finite difference schemes; accuracy and stability; ADI methods.

• Hyperbolic PDEs:
  Explicit finite difference methods; stability, accuracy and convergence; CFL condition; dissipation and dispersion; implicit methods.
  
  More advanced methods; discontinuous solutions, Rankine-Huginiot conditions, Riemann problems; finite volume schemes; TVD schemes.

• Finite element methods:
  Mathematical formulation and implementation of the Galerkin method for elliptic and parabolic equations.

• Boundary element methods:
  Reformulation of an elliptic PDE as an integral equation. Discretisation using simple boundary elements.

The course lectures will be given by the teaching and research staff of the Applied Mathematics and Operational Research group under the direction of Dr Stephen Shaw with the assistance of other colleagues.  External speakers may give lectures on specialist topics.

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