- 12:30 PM

Equation-Free and Equation-Assisted Computation for Spatially Distributed Multiscale Models

Liang Qiao, Department of Chemical Engineering,Princeton University, A-121 Engineering Quadrangle,Princeton University, Princeton, NJ 08544, Radek Erban, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, United Kingdom, Carl T. Kelley, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, and Yannis G. Kevrekidis, Princeton University, 6 Olden Street, Princeton, NJ 08544.

Deriving accurate macroscopic evolution equations (e.g. Partial Differential Equations) from detailed individual-based models is often a challenging task. Introducing closure assumptions -based on mathematics, or smart heuristics- can result in approximately valid macroscopic equations. In this work we illustrate how such "approximately correct" PDEs can be used to assist coarse-grained numerical computations based on the equation-free framework. We observe that the convergence of equation-free multiscale fixed-point solvers (based on, for example, Newton-GMRES) can be significantly accelerated through preconditioning which exploits approximate closures. Our model problem is a one-dimensional stochastic reaction-diffusion system that can exhibit Turing instabilities. We compute its coarse-grained bifurcation diagram using both equation-free and equation-assisted algorithms; stable as well as unstable coarse-grained, spatially varying solutions are computed and their (coarse-grained) stability is quantified. We also discuss possible extensions of the approach beyond this prototype example to more complex stochastic pattern formation problems.