Moments conservation in adaptive Vlasov solver

Abstract

We previously developed an adaptive semi-Lagrangian solver using a multiresolution analysis based on interpolets which are a kind of interpolating wavelets introduced by Deslauriers and Dubuc. This paper introduces a new multiresolution approximation for this solver which allows to conserve moments up to any order in the thresholding step by using the lifting method introduced by Sweldens. MSC 65Y05 82D10Key words: Vlasov; phase-space grid; adaptive; multiresolution; plasma physics; beam physics.The model we consider throughout this paper is the nonrelativistic Vlasov equationwhere the self electric field E is computed from Poisson's equations. The magnetic field is external and considered to be known.The numerical solution of the Vlasov equation is usually performed by particle-incell methods (PIC) which are known to suffer from numerical noise. To remedy this problem and obtain a more accurate description of the distribution function, methods discretizing the Vlasov equation on a mesh of phase space have been proposed [1,4,5]. In order to avoid the high numerical cost of such methods using a uniform and fixed mesh, we develop adaptive methods.Our adaptive method is overlaid on a classical semi-Lagrangian method. Adaptivity, that is refinement or derefinement of the mesh, is based on a multiresolution analysis (section 1, see [3] for more details). In the present work, we focus on moment conservation for the Vlasov equation during the thresholding step. This ensures in particular that the total number of particles is conserved. We explain (section 2) how to use the lifting procedure introduced by W. Sweldens [6] in order to ensure this conservation of moments. Relevant numerical results are presented in section 3.

A wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov–Maxwell system

“…These rules are derived heuristically. Although they are satisfactory in practice, these rules do not seem to be sufficient to prove estimate like the inequality (32). In [13], a more severe refinement rule is proposed: refine by q levels if 2 qðsÀ1Þ 6 jd e;j k j < 2 ðqþ1ÞðsÀ1Þ , with s the Hö lder smoothness of the underlying continuous wavelet system fw e k g ðk2Z;e2RÞ obtained as the limit of fw e;J k g ðk2Z;e2RÞ by letting J ?…”

“…The method was first conceived in [6] for the Vlasov-Poisson system, then used in the context of beam physics in [31,7], and parallelized with optimized data structure in [35]. In [32] the same method allows to conserve moments up to any order by using the lifting method introduced in [44].…”

Vlasov solver for longitudinal dynamics in beam delivery systems for x-ray free electron lasers

“…In other words, a beam with initial vanishing horizontal emittance will see its emittance grow as a consequence of collective forces. We assume that such an emittance growth will be sufficiently small that (17) remains close to the actual solution. If this is the case, we can obtain a reduced 2D Vlasov equation for the longitudinal motion by inserting (17) into (16) and integrating over the transverse variables x and p x :…”

“…A combination of the two might also be profitable. With regard to the first, recent work by Sonnendrücker and co-workers [17] using multiscale resolution and moving grids is promising, while grid-free methods proposed in [18] may be worthy of further exploration. Concerning the second strategy a change of variables to the ''interaction picture'' was proposed in [13].…”

Development of a 2D Vlasov Solver for Longitudinal BeamDynamics in Single-Pass Systems

“…A combination of the two might also be profitable. With regard to the first, recent work by Sonnendrücker and coworkers [14] using multiscale resolution and moving grids is promising, while grid-free methods proposed in [15] may be worthy of further exploration. Concerning the second strategy a change of variables to the "interaction picture" was proposed in [12].…”