William T. Taitano

A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth-Fokker-Planck equation

In this study, we demonstrate a fully implicit algorithm for the multi-species, multidimensional Rosenbluth-Fokker-Planck equation which is exactly mass-, momentum-, and energy-conserving, and which preserves positivity. Unlike most earlier studies, we base our development on the Rosenbluth (rather than Landau) form of the Fokker-Planck collision operator, which reduces complexity while allowing for an optimal fully implicit treatment. Our discrete conservation strategy employs nonlinear constraints that force the continuum symmetries of the collision operator to be satisfied upon discretization. We converge the resulting nonlinear system iteratively using Jacobian-free Newton-Krylov methods, effectively preconditioned with multigrid methods for efficiency. Single- and multi-species numerical examples demonstrate the advertised accuracy properties of the scheme, and the superior algorithmic performance of our approach. In particular, the discretization approach is numerically shown to be second-order accurate in time and velocity space and to exhibit manifestly positive entropy production. That is, H-theorem behavior is indicated for all the examples we have tested. The solution approach is demonstrated to scale optimally with respect to grid refinement (with CPU time growing linearly with the number of mesh points), and timestep (showing very weak dependence of CPU time with time-step size). As a result, the proposed algorithm delivers several orders-of-magnitude speedup vs. explicit algorithms.

Short note: Leveraging Anderson Acceleration for improved convergence of iterative solutions to transport systems

In this note we demonstrate that using Anderson Acceleration (AA) in place of a standard Picard iteration can not only increase the convergence rate but also make the iteration more robust for two transport applications. We also compare the convergence acceleration provided by AA to that provided by moment-based acceleration methods. Additionally, we demonstrate that those two acceleration methods can be used together in a nested fashion. We begin by describing the AA algorithm. At this point, we will describe two application problems, one from neutronics and one from plasma physics, on which we will apply AA. We provide computational results which highlight the benefits of using AA, namely that we can compute solutions using fewer function evaluations, larger time-steps, and achieve a more robust iteration.

Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov-Fokker-Planck-Ampère system, part I

In this study, we propose a charge, momentum, and energy conserving discretization for the 1D-1V Vlasov-Ampere system of equations on an Eulerian grid. The new conservative discretization is nonlinear in nature, but can be efficiently converged with a moment-based nonlinear accelerator algorithm. We demonstrate the conservation and convergence properties of the scheme with various numerical examples, including a multi-scale ion-acoustic shockwave problem.

Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov–Fokker–Planck–Ampère system, part II: Collisional aspects

In this study, we extend the moment-based acceleration algorithm for the charge, momentum, and energy conserving Vlasov–Ampere discretization developed in Ref. [1] by including a reduced Fokker–Planck operator. We propose an energy conserving discretization for the Fokker–Planck collision operator. We show by numerical experiment that the new algorithm 1) efficiently converges the nonlinearly coupled Vlasov–Fokker–Planck–Ampere system, and 2) accurately steps over stiff time-scales such as the inverse electron plasma frequency, and the electron–electron collision time-scale. We demonstrate that discrete energy conservation is critical to eliminate numerical heating issues when strong density gradients exist.

An equilibrium-preserving discretization for the nonlinear Rosenbluth–Fokker–Planck operator in arbitrary multi-dimensional geometry

The Fokker–Planck collision operator is an advection-diffusion operator which describe dynamical systems such as weakly coupled plasmas [1,2], photonics in high temperature environment [3,4], biological [5], and even social systems [6]. For plasmas in the continuum, the Fokker–Planck collision operator supports such important physical properties as conservation of number, momentum, and energy, as well as positivity. It also obeys the Boltzmann’s H-theorem [7–11], i.e., the operator increases the system entropy while simultaneously driving the distribution function towards a Maxwellian. In the discrete, when these properties are not ensured, numerical simulations can either fail catastrophically or suffer from significant numerical pollution [12,13]. There is strong emphasis in the literature on developing numerical techniques to solve the Fokker–Planck equation while preserving these properties [12–24]. In this short note, we focus on the analytical equilibrium preserving property, meaning that the Fokker–Planck collision operator vanishes when acting on an analytical Maxwellian distribution function. The equilibrium preservation property is especially important, for example, when one is attempting to capture subtle transport physics. Since transport arises from small O ( ) corrections to the equilibrium [25] (where is a small expansion parameter), numerical truncation error present in the equilibrium solution may dominate, overwhelming transport dynamics. Chang and Cooper [15] first proposed a discrete equilibrium preservation scheme for the Fokker–Planck operator in a 1D isotropic linear system. Larsen et al. [14] developed an equilibrium preserving scheme for Compton scattering in 1D by using the analytical expression for the transport coefficients. Buet and Le Thanh [19,20] developed a mass and energy conserving, positivity and equilibrium preserving scheme in the Landau (integral) formulation for a 1D isotropic Fokker–Planck operator by leveraging the natural (integral) structure of the Fokker–Planck collision operator. All of the approaches above

Plasma kinetic effects on interfacial mix

Mixing at interfaces in dense plasma media is a problem central to inertial confinement fusion and high energy density laboratory experiments. In this work, collisional particle-in-cell simulations are used to explore kinetic effects arising during the mixing of unmagnetized plasma media. Comparisons are made to the results of recent analytical theory in the small Knudsen number limit and while the bulk mixing properties of interfaces are in general agreement, some differences arise. In particular, “super-diffusive” behavior, large diffusion velocity, and large Knudsen number are observed in the low density regions of the species mixing fronts during the early evolution of a sharp interface prior to the transition to a slow diffusive process in the small-Knudsen-number limit predicted by analytical theory. A center-of-mass velocity profile develops as a result of the diffusion process and conservation of momentum.

Moment-Based Acceleration for Neutral Gas Kinetics with BGK Collision Operator

In this work, we present a moment-based accelerator algorithm for a Picard iteration applied to a neutral gas dynamics Boltzmann transport equation with a Bhatnagar-Gross-Krook collision operator. Traditional approaches relying on either explicit or Picard iteration schemes (i.e., source iteration) are severely limited for investigating time-scales much larger than the collisional relaxation time-scale, τ. We have developed a nonlinear accelerator algorithm that allows one to step over this stiff collision time scale and follow the hydrodynamic time scale of the problem when appropriate. The new algorithm relies on formulating a nonlinear, coupled system comprised of a high-order (HO) kinetic equation and a low-order (LO) fluid moment equation system. The HO equation provides self-consistent closures to the LO fluid equations, while the latter provides the required implicit-moment variables to evaluate the collision operator. We characterize the performance of the new algorithm on a Sod shock tube and a strong shock tube problem with varying Knudsen number.

Ion species stratification within strong shocks in two-ion plasmas

Strong collisional shocks in multi-ion plasmas are featured in many environments, with Inertial Confinement Fusion (ICF) experiments being one prominent example. Recent work [Keenan

Plasma ion stratification by weak planar shocks

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

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