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| Module Availability |
Autumn Semester
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| Assessment Pattern |
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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A test and a piece of assignment (including some programming work)
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25%
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2.5 hour closed-book examination
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75%
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Qualifying Condition(s)
A weighted aggregate mark of 50% is required to pass the module.
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| Module Overview |
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| Prerequisites/Co-requisites |
None |
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| Module Aims |
To introduce the concept of structure preserving numerical integration and the example of symplectic integrators for mechanical systems; To explain their good numerical behaviour using backward error analysis. |
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| Learning Outcomes |
By the end of the course the students should know how to design and implement simple symplectic Runge-Kutta methods and splitting methods. Moreover they should be able to explain the advantages of symplectic discretizations vs non-symplectic integrators using backward error analysis. |
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| Module Content |
Introduction:
Introduce examples of Hamiltonian systems from physics and their conserved quantities; motivate symplectic integrators versus non-symplectic integrators for some simple examples, in particular with respect to numerical conservation of energy.
One-step methods for ODEs:
Introduce consistency and order of convergence for one-step methods; treat some low order Runge-Kutta methods as examples.
Hamiltonian systems and symplectic discretizations:
Define symplectic maps, show that Hamiltonian flows are symplectic. Introduce implicit midpoint rule as symplectic integrator. Introduce simple partitioned symplectic Runge-Kutta methods (symplectic Euler method) and symplectic discretizations constructed by splitting methods; apply to problems from physics.
Backward error analysis for symplectic integrators:
Introduce the concept of a modified equation; derive exponential error estimates and approximate energy conservation for exponentially long times. Recall Noether’s theorem and prove conservation of angular momentum of symplectic methods.
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| Methods of Teaching/Learning |
Teaching is by lectures and tutorials. Learning takes place through lectures, tutorials, exercises and coursework. 3 hours lectures/tutorials/examples classes per week for 10 weeks. |
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| Selected Texts/Journals |
http://www.maths.surrey.ac.uk/personal/st/C.Wulff/Modules/MSM.GIN/msm.gin.html
A:
B. Leimkuhler and S. Reich. Simulating Hamiltonian Dynamics.
Cambridge Monographs on Applied and Computational Mathematics, 14.
ISBN – 0521772907, 2004.
B:
E. Hairer and C. Lubich and G. Wanner. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Springer Verlag, Berlin, 2002
C:
J. M. Sanz-Serna and M. P. Calvo. Numerical Hamiltonian problems. Chapman and Hall, London, 1994. |
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| Last Updated |
30 July 2007 |
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