For more information about this meeting, contact Yakov Pesin, Ping Xu.
|Title:||General Techniques for Constructing Variational Integrators|
|Seminar:||Department of Mathematics Colloquium|
|Speaker:||Melvin Leok, UCSD|
|Geometric numerical integrators are numerical methods that preserve the geometric structure of a continuous dynamical system, and variational integrators provide a systematic framework for constructing numerical integrators that preserve the symplectic structure and momentum, of Lagrangian and Hamiltonian systems, while exhibiting good energy stability for exponentially long times.
The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi's solution of the Hamilton--Jacobi equation. These two characterizations lead to the Galerkin and shooting-based constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We will relate properties of the quadrature formula, finite-dimensional function space, and underlying one-step method used with the properties of the associated variational integrators.
This approach can also be generalized to the case of degenerate Hamiltonian systems, which allows for the construction of variational integrators for Hamiltonian systems which do not have a Lagrangian analogue. When extended to the case of Hamiltonian field theories, this also provides a systematic framework for constructing geometric integrators that are automatically multisymplectic.|
Room Reservation Information
|Date:||11 / 10 / 2011|
|Time:||04:00pm - 05:00pm|