| Computer-assisted investigations are a valuable tool in understanding the behavior of complex systems. We aim to increase the efficiency and range of such (both microscopic and macroscopic) investigations, by developing novel ways to identify such systems' essential components. Reaction-diffusion equations are Partial Differential Equations (PDEs) whose function is to model macroscopically a broad class of systems exhibiting chemical (reaction) and collisional (diffusion) interactions. As such, their successful analysis and efficient numerical simulation has far-reaching consequences in the biological, chemical, and physical context. Realistic reaction-diffusion models often involve large numbers of unknowns; for many systems of interest, this number is into the hundreds. Additionally, the system dynamics occur typically in multiple time scales, with relatively few modes remaining active over long time periods (stiffness). These characteristics obscure the essential dynamics and make numerical computations time-consuming. On the other hand, stiffness causes solutions to cluster on subsets (slow manifolds) of the solution space, effectively reducing complexity. The identification of slow manifolds allows one to bypass initial fast transients, thus removing stiffness and speeding up computations. This way, simplified models retaining the essential system characteristics can be obtained. It is this identification that forms the subject of reduction. Nearly every reduction method is developed on an intuitive basis. Additionally, although a multitude of methods are employed to reduce non-diffusive systems (described by Ordinary Differential Equations (ODEs)), PDE reduction relies largely on ad hoc techniques. In short, PDE reduction methods are scarce and their premises often of ambiguous validity. Recently, the Equation-Free (EF) approach to multiscale numerical simulations was developed to bridge computationally the microscopic and macroscopic levels of description. This approach uses computational models of microscopic ensembles to derive macroscopic information, rather than explicitly available (macroscopic) reaction-diffusion PDEs. The aim of the proposed research is twofold. First, to develop reduction methods for PDEs. Second, to develop reduction methods implementable in EF settings. If successful, this research will provide a powerful tool to the efficient numerical investigation of reaction-diffusion equations. |
