An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes

Abstract

In this study, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth-Fokker-Planck (RFP) equation, which is exactly mass-, momentum-, and energyconserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local-thermal velocity, v th (t), and explicitly considers the resulting inertial terms in the Fokker-Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker-Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic v th-ratio-based expansion of the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. In particular, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.

An adaptive, implicit, conservative, 1D-2V multi-species Vlasov–Fokker–Planck multi-scale solver in planar geometry

“…We lag the time level between the BDF2 coefficients and the normalization velocity for well-posedness of the velocity-space grid motion [13]. The term d corresponds to the inertial term due to the spatial variation of the normalization velocity, v * α , with…”

“…Finally, we combine all the conservation properties. We remark, that the detailed derivations of conservation symmetries for the temporal terms in the Vlasov equation and for the collision operator have been considered elsewhere [14,13], with a more numerically robust generalization based on a constrained-minimization approach discussed in App. Appendix D, and, therefore, only the spatial gradient terms are considered here.…”

“…Similarly to Ref. [13], we re-write the expression in (3.25) by multiplying by v * and using the chain rule to obtain…”

“…In this study, we extend the conservative, multiple-dynamic velocity-space adaptive strategy developed in Ref. [13] to a spatially inhomogeneous, 1D Cartesian system. We consider a quasi-neutral plasma with multiple kinetic ion species and fluid electrons.…”

A discontinuous Galerkin fast spectral method for the multi-species Boltzmann equation

“…Possible remedies include adaptive mesh in velocity space (cf. [29]), using an asymptotic model valid for large mass ratios (cf. [30]), or introducing independent velocity grid for each species wherein different collision types for every (i, j) pair are treated independently (cf.…”