MA 273A Advanced Techniques in Computational Mathematics - Geometric Numerical Integration

MWF 1:00-1:50pm, APM 2402.

Prof. Melvin Leok
Office: APM 5763
Email: mleok @ math . ucsd . edu
Office Hours: TBA.

Course Handout
A short video introduction to my research.
The resources below are password protected with the user name ma273a, and the password is the first 4 digits of:
$\sqrt{6\sum_{n=1}^\infty}{1}{n^2}}$ .
Introductory Lecture
Course Notes

The final presentations will be held on Monday, December 7, between 11:30am and 2:30pm, in APM 2402. The schedule is as follows:

Time Slot	Student	Title
11:30am-11:50am	Eric Arobone	Lagrangian Analysis of Point Vortices and Coherent Structures in Stratified Turbulence
11:55am-12:15pm	Slava Kungurtsev	Variational Integration of the Schrodinger Equation
12:20pm-12:40pm	David Lipshutz	Composition and Symplectic Splitting Methods for Gaussian Wavepacket Dynamics
12:40pm-1:20pm	Break
1:20pm-1:40pm	Maximillian Metti	Method of Lines for Hamiltonian PDEs
1:45pm-2:05pm	Hubert Nguyen	Symplectic Integrators for Classical Spin Systems
2:10pm-2:30pm	David Garber	Highly Oscillatory Problems

This course is relevant to engineers, scientists, and mathematicians with an interest in long-time simulations of mechanical systems, including applications to robotic motion planning, astrodynamics, rigid-body, molecular and stellar dynamics. The application areas addressed will be tailored to the interests of the course participants.

Many differential equations of interest in the physical sciences and engineering exhibit geometric properties that are preserved by the dynamics. Recently, there has been a trend towards the construction of numerical schemes that preserve as many of these geometric invariants as possible.
Such methods are of particular interest when simulating mechanical systems that arise from Lagrangian or Hamiltonian mechanics, wherein the preservation of physical invariants such as the energy, momentum, and symplectic form can be important when simulating long-time dynamics of such systems.
In applications arising from astrodynamics and robotics, the dynamics evolve on nonlinear manifolds such as Lie groups, and in particular the rotation group, and the special Euclidean group. Numerical schemes that respect the underlying nonlinear manifold structure will also be discussed.
This course will begin with an overview of classical numerical integration schemes, and their analysis, followed by a more in depth discussion of the various geometric properties that are of importance in many practical applications, followed by a survey of the various geometric integration schemes that have been developed in recent years. Issues pertaining to the analysis and implementation of such schemes will also be addressed.

A strong undergraduate background in linear algebra, and differential equations; Some familiarity with numerical methods, classical mechanics and differential geometry would be helpful, but not essential; Programming experience in any language, e.g., C/C++, FORTRAN, MATLAB.

Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics
Leimkuhler, Reich, Cambridge University Press, 2005. ISBN: 0521772907 [ Electronic Version ]

Geometric Numerical Integration, Springer Series in Computational Mathematics
Hairer, Lubich, Wanner, 2nd Edition, Springer-Verlag, 2006. ISBN: 3540306633 [ Electronic Version ]
Geometric numerical integration illustrated by the Störmer-Verlet method, Hairer, Lubich, Wanner, Acta Numerica, 399-450, 2003.
Discrete mechanics and variational integrators, Marsden, West, Acta Numerica, 357-514, 2001.
Asynchronous Variational Integrators, Lew, Marsden, Ortiz, West, Arch. Rational Mech. Anal., 167(2), 85-146, 2003.
Splitting methods, McLachlan, Quispel, Acta Numerica, 341-434, 2002.
Linear algebra algorithms as dynamical systems, Chu, Acta Numerica, 1-86, 2008.
Lie-group methods, Iserles, Munthe-Kaas, Norsett, Zanna, Acta Numerica, 1-148, 2005.
Manifolds, Tensor Analysis, and Applications, Marsden, Ratiu, Third Edition, Electronic Version.
Introduction to Mechanics and Symmetry, Marsden, Ratiu, Second Edition, Springer-Verlag, 2002.
Chapter 9 (An Introduction to Lie Groups), Mechanics and Symmetry, Second Edition, Electronic Version.

Your grade in the course is based on your project and your 20-minute project presentation during the time of the final.
The topic of your project should be decided in consultation with the instructor before the end of the first month of class.