Multiscale Techniques for Nonlinear Dynamical Systems: Applications and Theory

Abstract

Most interesting real world systems can be understood at multiple scales of detail. A physical system such as a closed container of gas particles can be understood in terms of hydrodynamic flows, molecules and atoms exerting forces upon one another, or evolving wavefunctions for each component particle. A multiscale approach can be used to understand the interplay of different scales of detail in problems that lack time or spatial scale separation. We present the Mori-Zwanzig formalism as a general framework for understanding multiscale methods. Another popular multiscale method, the heterogeneous multiscale method, is shown to be a special case of this framework. The heterogeneous multiscale method framework is applied to a plasma physics problem. We then derive a new multiscale scheme from the Mori-Zwanzig formalism called the complete memory approximation, and apply it to the Korteweg-de Vries equation, the 3D Euler's equations, and Burgers' equation. Surprising scaling results shed light into the complex role played by memory in reduced order models of partial differential equations.