Wavelet Techniques for Coarse Projective Integration in Multiscale Plasma Dynamics

Introduction Multi-scale problems such as magnetic reconnection and turbulence are notoriously hard to simulate because the physics of micro and macroscales are strongly linked. Over the past few years, a simulation framework called Equation-Free Projective Integration (EFPI) has been applied to a variety of multi-scale phenomena in engineering problems in which coarse-scale behavior can be obtained through short-time simulations within the fine-scale models (microscopic, stochastic etc) [1]. Recently, the first application of EFPI to a plasma system has been developed and implemented by Shay ([2]). He studies the propagation and steepening of a 1D ion acoustic wave using both a kinetic particle in cell (PIC) code as well as an EFPI code. He finds that the EFPI code reproduces the PIC code well, however differences arise due to physics assumptions made in the lifting part of the algorithm, specifically that the ion velocity probability density functions (PDF) remain Maxwellian and that the plasma remains quasineutral. This paper discusses a generalization of Shay’s projective integration scheme that removes the assumption on the velocity PDF. In particular, we propose a scheme that estimates the joint x-v phase space PDF using non-linear wavelet approximation. We use a small number of wavelet coefficients, which represent the coarse grained structure of the joint PDF, as the macroscopic observables of the evolving system. An alternative method for tracking of an arbitrary PDF has been proposed by Zou ([3]). They use projection of the inverse Cumulative Distribution Function onto an orthogonal polynomial basis (shifted Legendre polynomials) to obtain relevant EFPI quantities. We believe that wavelets, with their inherent multiresolution structure, and localized behaviour, are better suited for representing arbitrary PDF’s in the context of EFPI for plasma dynamics simulations.