Mihail E. Kavousanakis, School of Chemical Engineering, National Technical University of Athens, Athens, Greece, Andreas G. Boudouvis, School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou St., Zografos, Athens, GR-157 80, Greece, and Yannis G. Kevrekidis, Princeton University, 6 Olden Street, Princeton, NJ 08544.
Projective and coarse projective integration are numerical methods for the accelerated computation of the time evolution of multiscale systems. In this work, we study systems with continuous symmetries (e.g. translational invariance, associated with traveling solutions and scale invariance associated with self-similar solutions). We argue that the best results for the projective and coarse projective integration methods are obtained when the computation is performed in a “co-evolving” frame, i.e. the frame which is co-traveling, co-collapsing, co-exploding or co-rotating with the evolving solution.
We illustrate the theoretical ideas on the one-dimensional translationally invariant FitzHugh-Nagumo (FHN) PDE. We also construct an individual-based kinetic Monte Carlo model motivated from the FHN kinetics on which the coarse-grained version of the approach is illustrated. The coarsely scale invariant system we use is the one-dimensional diffusion of particles. Again, we present the efficiency of projective integration in the co-evolving frame for both the macroscopic diffusion PDE and for a random-walker particle based model.