This module is not currently running in 2024 to 2025.
The equations studied in this module will be ordinary differential systems, especially Hamiltonian systems. The aim of this subject area is to obtain and study numerical solutions of these systems that preserve specific qualitative and geometric properties. For certain differential equations, these geometric methods can be far superior to standard numerical methods. The syllabus includes: A review of basic numerical methods, variational methods and Hamiltonian mechanics; Properties that numerical methods can preserve (first integrals, symplecticity, time reversibility); Geometric numerical methods (modified Euler and Runge-Kutta methods, splitting methods); Use and misuse of the various notions of error.
Total contact hours: 30
Private study hours: 120
Total study hours: 150
70% Examination, 15% Coursework, 15% Project
All texts are available in the Templeman library and are recommended for background reading.
Books:
Simulating Hamiltonian Dynamics, Leimkuhler and Reich, Cambridge University Press, 2005.
Geometric Numerical Integration, Hairer and Lubich and Wanner, second edition, Springer Verlag, 2006.
Review articles:
Six Lectures in Geometric Integration, MacLachlan and Quispel, in Foundations of Computational Mathematics pages 155-210, ed. R. DeVore, A. Iserles, E. S¨uli, Cambridge University
Press, Cambridge, 2001. (Available online)
Geometric Integration and its Applications, Handbook of Numerical Analysis, Volume XI NorthHolland 2000.
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successfully completing the module students will be able to:
1 derive numerical methods and their properties;
2 demonstrate appreciation of the geometric interpretation of differential equations and numerical algorithms;
3 demonstrate understanding of the meaning and interpretation of error in approximations, in particular the relative importance of local errors versus global properties;
4 demonstrate appreciation of the importance, meaning and interpretation of numerical stability;
5 apply specific sophisticated numerical tools which preserve certain mathematical structures;
6 use mathematical software such as MatLab to masters level.
The intended generic learning outcomes. On successfully completing the module students will be able to:
1 reason and deduce confidently from given definitions and constructions;
2 show an enhanced understanding of what is meant by an answer to a modelling problem;
3 read independently and manage their time;
4 demonstrate enhanced skills with mathematical and graphical software, to postgraduate level;
5 show their matured problem formulating and solving skills;
6 apply a wide variety of Calculus, Linear Algebra, Mathematical Modelling, and Mathematical Methods based skills.
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