Solving Partial Differential Equations with Chaotic Asynchronous Schemes in Multi-Interaction Systems

Abstract

Within the framework of multi-interaction systems (MIS), we aim at proposing algorithms for solving some partial differential equations (PDE), that are commonly used for modeling phenomena like transport or diffusion, as they often occur in a complex system involving natural phenomena. Unlike classical synchronous ones, schemes for MIS have to be compatible with chaotic asynchronous iterations, which enable multi-model/scale, interactive and real-time simulations. In our context, the notion of asynchronous iteration expresses the fact that activities, each modeling a phenomenon, have their own lifetime and are processed one after the other. These activations are processed by cycles, in a random order -to avoid computation bias-, what we name chaotic iterations. We provide MIS-compatible schemes to simulate transport phenomena, thermal diffusion phenomena and the spreading phenomenon of a wave packet. Our schemes are based on interactions that represent sorts of Maxwell daemons: transfers of flows between several separate environments given by a spatial resolution grid. We establish formal proofs of convergence for our transport methods. We experiment an efficient asynchronous diffusion scheme, and couple both schemes for solving the advectiondiffusion problem. We finally illustrate a multi-interaction method for the spreading of a wave packet described by the Schrodinger equation. Results are compared to classical numerical methods and they show that our methods are as accurate as classical ones, whilst respecting MIS constraints.