Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
There are important systems that need to be modeled on relatively long time or space scales, but the dynamics can only be advanced on short time or space scales. These systems with multiple, separated scales include coupling molecular dynamics with the macroscale behavior of a material, or using a Boltzmann particle model to predict large scale patterns in a fluid flow. Patch Dynamics is an efficient approach to bridge these scales by using locally averaged properties of the short time and small spatial scales to advance and predict the dynamics of the long time and space scale dynamics.
The figure illustrates patch dynamics for a one-dimensional physical system where the microscale variable u(x) varies rapidly, but the average of u(x) varies slowly. The boundary conditions for the patches are defined by extending the microscale solution into a buffer region surrounding each patch. The patches communicate with each other via boundary conditions similar to the way finite difference approximations of partial differential equations communicate to the surrounding grid points.
