Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmas

Asymptotic-preserving schemes for multiscale physical problems

“…Further, a class of semiimplicit schemes has been proposed (Filbet and Rodrigues 2016, 2017, Hairer, Lubich and Wang 2020, Ricketson and Chacón 2020 to capture accurately the non-stiff part of the evolution while allowing for coarse discretization parameters. It allows us to capture the asymptotic limit of the two-dimensional Vlasov-Poisson system with a uniform magnetic field Rodrigues 2016, Filbet, Rodrigues andZakerzadeh 2021). In many respects these schemes are remarkably natural and simple, and can be adapted to toroidal geometry as in Filbet and Rodrigues (2020).…”

Asymptotically Preserving Particle Methods for Strongly Magnetized Plasmas in a Torus

“…In this regime, the time step can be subject to stability constraints related to the smallness of Larmor radius. To avoid this limitation, our approach is based on higher-order semi-implicit numerical schemes already validated on dissipative systems [3] and for magnetic fields pointing in a fixed direction [9,10,12]. It hinges on asymptotic insights gained in [11] at the continuous level.…”

“…Other approaches are based on the construction of efficient particle solvers for the original dynamics, to be used as a piece of a PIC scheme. Over the last decade, considerable efforts have been devoted to the design of such solvers and we refer the reader to [2,23,8,26,9,10,13,5,6,22,14,24,12,25,15,7] for both significant contributions and relevant entering gates to the now abundant literature. Along these years, roughly speaking, two kind of goals were assigned to the built numerical schemes.…”

“…The class of semi-implicit schemes proposed in [9,10,12] focuses on the less ambitious goal of guaranteeing only the asymptotic preserving property, thus the recovering of both the exact dynamics as ∆t goes to zero, uniformly with respect to ε, and the slow reduced dynamics, including the guiding-center particle motion, when ε goes to zero, uniformly with respect to ∆t. We stress that by many respects those schemes are remarkably natural and simple, and may easily be inserted in a standard PIC code.…”

“…For complex geometries, our schemes are designed as high-order semi-implicit schemes [3] applied to an augmented formulation. Our supporting strategy is quite systematic and versatile, but so far we have implemented it only for homogeneous magnetic fields [9,12] and magnetic fields pointing in a fixed direction [10].…”