(561a) Equation-Free and Equation-Assisted Computation for Spatially Distributed Multiscale Models

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.