Computational Methods for Ordinary Differential Equations

29 October - 02 November 2007

Aims

To introduce modern numerical methods for ordinary differential equations (ODEs) and their use in scientific computation

Objectives

By the end of this module students should be able to:

• Discuss well-posedness of ODEs
• Understand how to determine the local truncation error of an ODE solver
• Understand how to derive Taylor series methods, coefficients of explicit Runge-Kutta and explicit/implicit linear multistep schemes.
• Examine the behaviour of parasitic solutions and analyse stability of studied numerical methods.
• Solve ODEs via library software by making a sensible choice of algorithms and know of approaches to qualitatively assess the resulting solution.

Syllabus

• Examples of ODEs; first, second and higher order ODEs; conversion to a system of first order equations; some analytic methods of solution; well-posedness; introduction to differential algebraic equations (DAEs)
• Numerical Methods for ODEs: Single Step Methods: Taylor series and Euler's methods; numerical errors: local and global truncation errors; stability; Runge-Kutta methods; error control
• Linear Multi-Step Methods: Adams-Bashforth, Adams-Moulton; error control; stability
• Numerical Methods for Stiff Systems of ODEs: stability and convergence of ODE solvers; parasitic solutions; examples; BDF and implicit Runge-Kutta methods; consequences of stability restrictions.
• Software for ODEs: numerical libraries and packages for ODEs and DAEs.
• Solution of Large Scale ODEs: sparse finite-difference; automatic differentiation; Jacobian compression
• Method-of-Lines for PDEs: solution of time-dependent PDEs by ODE solvers via Method-of-Lines discretisation

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 Shaun Forth with the assistance of other colleagues.  External speakers may give lectures on specialist topics.