Numerics for nonlocal potentials and highly-oscillatory PDEs

Nonlocal interactions are ubiquitous in (quantum) physic, their numerical evaluation requiring both accuracy and efficiency. Typically they have convolution integral structure, like the Newtonian potential 1/x. The project aims at developing efficient and accurate numerical algorithms for the evaluation of nonlocal potentials and investigating the numerics of high-order averaging schemes for highly oscillatory system, e.g. strongly confined systems of quantum particles involving some longrange interactions that are modeled by nonlocal potentials. The project partners are Leslie Greengard (Courant Inst., NYU), Shidong Jiang (NJIT New Jersey), Florian Méhats (IRMAR at U. Rennes) and Norbert J. Mauser (WPI at U. Wien). Especially for strongly confined system great numerical challenges occur since the pronounced anisotropy induces highly oscillatory behavior in space and time. Here the expertise of Méhats and Mauser is helpful. The singularity of convolution interaction kernel needs to be dealt with carefully, and also the (slow decay) long-range effect demands appropriate truncation. We develop such methods and their rigorous numerical analysis. The nonuniform Fast Fourier Transform (NUFFT) and Gaussian-Sum (GS) methods, recently introduced and developed further by the PI and the project partners, are state-of-the-art solvers. Together with Greengard, Jiang and Mauser these methods will be improved, extended and implemented. Fast oscillations usually impose severe step-size restriction so as to capture the long-time dynamics of such stiff systems. So far, many numerical schemes, based on asymptotic analysis, have been developed in order to alleviate the formidable computational cost imposed by the fast oscillations. Geometric integrators, including stroboscopic averaging method (SAM) and multi-revolution composition methods (MRCM) are promising candidates for general highly oscillatory systems, e.g. strongly confined kinetic equations. Here the expertise of the pioneers Méhats and Philippe Chartier in Rennes is crucial. Moreover, for strongly confined systems that involve nonlocal potential, both problems occur simultaneously. Major project goals are: Analytical and Numerical study of the two-component rotating dipolar BEC. Comprehensive study of nonlocal potential solvers, with further extensions. High-order averaging method for kinetic equations. Analysis and numerics for confined quantum systems with magnetic fields. 1

No additional funding sources recorded.