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William, T Taitano
Moment-Based Acceleration for a Collisional Kinetic Plasma Model

Theoretical Division
MS B216
Los Alamos National Laboratory
Los Alamos
NM 87545
taitano@lanl.gov
Luis Chacon
Dana Knoll

When one considers collisional kinetic plasmas with charge-separation, disparate separation in length and time scales arise. For the time-scales, many orders of magnitude separation exist between the inverse electron plasma frequency, collision time-scales, and the simulation time. Similarly, for the length-scales, there exist many orders of magnitude separation between the Debye length, collisional mean-free-path, and the system length. If one is interested in studying system time and length scales, explicit integration methods do not suffice. Explicit approaches are required to respect the CFL stability constraint, which imposes severe limitations on the time-step size that can be used. Implicit methods can effectively use a much larger time-step size, but require solving a nonlinear problem. Fixed point iteration methods such as Picard are simple to implement, but can be slow to converge when stepping over fast electron time-scales (e.g. inverse electron plasma frequency), and can diverge in certain cases [1]. Moment-based accelerations have been demonstrated in various applications [2,3,4,5] to significantly improve both the convergence and the robustness of the Picard iteration algorithm. Moment-based acceleration algorithms employ low-order (LO) moment-field equations (PDE), which live in a reduced space (x, t), as a coarse-grid accelerator for the high-order (HO) integro-differential kinetic equations, which live in the phase-space (x, v, t).

In this work, we develop a HOLO moment-based nonlinear convergence acceleration method for a collisional plasma model with charge-separation effects. The model is described by the Vlasov-Fokker-Planck-Ampère (VFPA) system. When one considers long-time integration of collisional plasmas with charge-separation, preserving discrete conservation properties become an important accuracy concern. In particular, charge, and energy conservation are critical properties that must be conserved rigorously to prevent issues with numerical charge-separation and numerical heating. Standard discretizations of the VFPA equation are capable of preserving one of these properties, but not both. In order to simultaneously preserve both conservation properties, a new nonlinear discretization scheme is developed. The new discretization scheme is based on introducing and satisfying a set of discrete nonlinear constraints, which require nonlinear convergence to be enforced. In order to achieve nonlinear convergence acceleration, the continuity (0th moment), momentum (1st moment), and energy (2nd moment) equations must be included in the LO system, with stiff electron thermal conduction physics exposed. We use a multi-species, multi-length, and time scale model problem to demonstrate that the proposed algorithm can employ time-step sizes much larger than fast electron time-scales, yet obtain the solution very efficiently. Additionally, we show that the nonlinearity in the new conservative discretization scheme can be dealt with effectively through the HOLO moment acceleration algorithm.

[1] J.A. Willert, et al., J. Comp. Phys., submitted (2013).

[2] D.A. Knoll et al., Nuc. Sci. and Engn., Vol. 167 (2011).

[3] H.K. Park et al., Tranps. Theory and Stat. Phys., Vol. 41 (2012).

[4] W.T. Taitano et al., SIAM J. Sci. Comp., Vol 35 (2013).

[5] W.T. Taitano et al., Transp. Theory and Stat. Phys., submitted (2013).




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Copper Mountain 2014-02-23