Plasma Seminar

Dr. Luis Chacon, Los Alamos National Laboratory

"Implicit, conservative PIC algorithms for multiscale collision less plasma simulation"

4:00pm, RLM 11.204

Abstract: Collisionless plasmas are described by the Vlasov-Maxwell equations. This set of equations is high-dimensional (3D+3V+time), highly nonlinear, and remarkably multi-scale, supporting disparate time and length scales. These features make its efficient numerical integration extremely challenging.

The high-dimensionality of these equations have made particle methods (PIC) quite attractive. PIC is naturally adaptive in velocity space, but can be noisy, and is generally problematic for long-term integrations of the Vlasov-Maxwell equations due to both accuracy limitations (e.g., lack of energy conservation results in secular energy growth which subtracts fidelity from the simulation) and efficiency ones (particle methods are typically explicit, and feature both temporal and spatial stability constraints that force the resolution of both the fastest frequencies and the smallest length-scales supported by the model). These limitations make the long-term, system-scale PIC simulation of physical systems challenging, even with the most powerful supercomputers.

Recently, fully implicit, nonlinear algorithms have been proposed for both electrostatic [1,2] and electromagnetic [3,4,5] PIC descriptions that enable for the first time truly multiscale kinetic simulations of collisionless plasmas with particle methods. These algorithms 1) feature exact conservation properties, thus avoiding secular growth of conserved quantities, and 2) ameliorate the numerical stability constraints of explicit PIC, thus allowing the use of large time steps and mesh sizes (compatible with the physics of interest). The approaches, based on modern nonlinear iterative methods, minimize the solver memory footprint by nonlinearly enslaving the kinetic component (particles) to the field equations. Thus, only field equations appear explicitly in the nonlinear residual, resulting in only modest memory requirements for the nonlinear solver. Only a single copy of the particles is needed, as with explicit implementations.

However, despite drastically decreasing the number of degrees of freedom (by allowing larger mesh cells) and the number of time steps required for a given simulation (by allowing large time steps), the resulting algebraic system remains extremely ill-conditioned and requires effective preconditioning for efficiency. A powerful advantage of nonlinear kinetic enslavement is that one can explore moment-based preconditioners. In this talk, we will introduce the fully implicit, conservative multidimensional PIC method, and our approach to fluid preconditioning. We demonstrate the promise of the approach with various challenging numerical examples.

REFERENCES

[1] G. Chen, L. Chacón, D. C. Barnes, “An energy- and charge-conserving, implicit, electrostatic particle-incell algorithm,” J. Comput. Phys., 230, 7018–7036 (2011)
[2] G. Chen, L. Chacón, C. Leibs, D. A. Knoll, W. Taitano, “Fluid preconditioning for Newton-Krylov-based, fully implicit, electrostatic particle-in-cell simulations”, J. Comput. Phys., 258, p. 555–567 (2014)
[3] G. Chen, L. Chacón, “An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm,” Comput. Phys. Commun., 185 (10), 2391-2402 (2014)
[4] G. Chen, L. Chacón, “An energy- and charge-conserving, nonlinearly implicit, multidimensional electromagnetic Vlasov–Darwin particle-in-cell algorithm,” Comput. Phys. Commun., 197, 73-87 (2015)
[5] L. Chacón, G. Chen, “A curvilinear, fully implicit, conservative electromagnetic PIC algorithm in multiple dimensions,” J. Comput. Phys., 316, 578–597 (2016)