Magneto-Fluid Dynamics Seminar
Model reduction and simulation design for Hamiltonian systems
Speaker: Joshua Burby, Courant Institute, NYU
Location: Warren Weaver Hall 905
Date: Tuesday, December 12, 2017, 11 a.m.
Synopsis:
Hamiltonian systems, both finite- and infinite-dimensional, are ubiquitous in the physical sciences. In plasma science in particular, these systems tend to involve a variety of spatiotemporal scales that interact both nonlinearly and nonlocally. It is therefore an important task to develop model reduction and simulation strategies that are adapted to both Hamiltonian mechanics and multi-scale physics. In order to describe a new approach to addressing these challenges, I will discuss the synergistic notions of variational slow manifold reduction and symplectic rectification. Variational slow manifold reduction, which is a systematic model reduction tool for fast-slow Hamiltonian systems, produces simplified Hamiltonian systems that describe "balanced" system motions, i.e. trajectories that do not contain the fastest timescale in the problem. Symplectic rectification is a straightforward application of Moser's trick that transforms a complicated symplectic form into a nearby symplectic form that is easier to study. The careful application of symplectic rectification significantly simplifies the development of symplectic integrators, which have proven to be invaluable tools for the direct simulation of Hamiltonian systems with multiple time scales. After summarizing several recent results in the modeling and simulation of magnetized plasmas that this pair of tools helped to uncover, I will sketch a picture of how refinements of these tools may be useful in solving a variety of more challenging problems. The latter include developing reduced Hamiltonian models for systems with multiple space scales, renormalization of slow balanced dynamics by non-resonant normal oscillations in fast-slow Hamiltonian systems, and the development of continuum (e.g. Galerkin) symplectic integrators for weakly-nonlinear fluid and kinetic models.