The Moonrise project aims at exploring modeling, mathematical and numerical issues originating from the presence of high oscillations in nonlinear PDEs mainly from the physics of nanotechnologies and from the physics of plasmas.

Simulating numerically a fast-oscillating phenomenon usually imposes severe step-size restrictions in order to correctly capture the stiff dynamics. In many occurences of high oscillations, ad-hoc numerical methods have been developed — with partial success — which aim at capturing the long-time dynamics of the solutions in contrast with standard techniques which are enslaved to follow oscillations at a formidable computational cost. At the other end of the spectrum, several mathematical techniques aim at describing the solutions in their asymptotic limit, i.e. when a small parameter (which could be the inverse of a frequency or a normalized Planck's constant) tends to zero. Instead of having sophisticated tools of analysis on the one hand, and heuristic numerical methods on the other hand, this project aspires to accommodate the two and to provide not only fruitful high-order asymptotic models for the purpose of mathematical analysis but also efficient numerical methods derived from them.

Our target models are the following ones:

— Highly oscillatory regimes in nanoscale physics, including nonlinear Schrödinger equations for confined quantum systems (electrons in nanostructures, Bose-Einstein condensates, transport in graphene), multiscale physics with semiclassical or nonrelativistic scalings.

— Charged particles in strong magnetic fields, with models for strongly magnetized plasmas in Tokamaks devices or for space plasmas (the earth's magnetopause). Here, the models are Euler or Vlasov equations coupled to Maxwell's equations or, more simply, the Poisson equation.

— Quasi-neutral regimes for kinetic, fluid or diffusive models. These models are of great interest for the modeling of tokamaks and space plasmas or corrosion of iron-based alloys in nuclear waste repositories.

Our objective is to design reduced models and suitable efficient numerical schemes, for which the time steps will not be constrained by fast oscillations. In particular, we will develop strategies to construct uniformly accurate numerical schemes with respect to the highest frequencies of the phenomenon. These schemes will be assessed on configurations of increasing complexity: nanoscale devices in two or three dimensions and plasma devices up to Tokamak configurations. We will combine tools from asymptotic analysis, geometric numerical integration, high order averaging techniques and asymptotic-preserving numerical methods.