I will be offering a graduate special topics course on geometric numerical integration in Spring 2008. Students who are interested are strongly encouraged to preregister for the course before the registration deadline of November 1, 2007, in order to ensure that the course has sufficient enrollment to be offered.

Course Description

MATH 692: Geometric Numerical Integration
Spring 2008, TTh 12pm-1:15pm

Target Audience:
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.

Course Description:
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.

Background:
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.

Instructor:
Prof. Melvin Leok, Department of Mathematics
Office: Math 430, phone: 49-63578, email: mleok@math.purdue.edu
http://www.math.purdue.edu/~mleok/

Textbook:
Simulating Hamiltonian Dynamics
Cambridge Monographs on Applied and Computational Mathematics),
Leimkuhler, Reich, Cambridge University Press, 2005. ISBN: 0521772907

Course Topics:
Numerical methods, Hamiltonian mechanics, Geometric integrators, Modified equations, Higher-order methods, Constrained mechanical systems, Rigid body dynamics, Adaptive geometric integrators, Highly oscillatory problems, Molecular dynamics, Hamiltonian PDEs.

Grading:
40 % — 6 Theoretical homework assignments (1 week assignments)
30 % — 3 Numerical implementation projects (2 week assignments)
30 % — 1 Final project involving a non-trivial application of geometric integrators.