Implicit particle simulation of electromagnetic plasma phenomena

We apply the second-order formulation of Maxwell's equations proposed by Jiang et al. (1996, J. Comput. Phys. 125, 104) to the solution of the implicit formulation of the three-dimensional, time-dependent Vlasov-Maxwell's system. An implicit finite difference algorithm is developed to solve the Maxwell's equations in a bounded domain with physical boundary conditions comprising electrically conducting walls (perfect conductors) and constant magnetic flux walls. We formulate the boundary conditions for Maxwell's equations to satisfy Poisson's equation throughout the domain by solving it only on the boundary. This eliminates the need for a separate projection step. We compare numerical results with analytical solutions for electromagnetic waves in vacuo, and using the implicit particle-in-cell code CELESTE3D, we test the new solver on the geospace environment modeling magnetic reconnection challenge problem. c 2002 Elsevier Science (USA)

show abstract

“…Kamimura et al solve Eq. (43) [27]. On the other hand, Hewett and Langdon find that a formulation consistent with Eq.…”

Section: Temporal Discretizationmentioning

confidence: 98%

A Simplified Implicit Maxwell Solver

Ricci

Lapenta

Brackbill

2002

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…We proceed to outline the derivation of the local error estimate for the particle orbit equations, required for Eq. (19) in the main text. In what follows, we omit the subscript "p" in particle quantities for convenience.…”

Section: Appendix B Local Error Estimate Of Particle Orbit Integratormentioning

confidence: 99%

“…This realization drove the exploration of implicit PIC starting in the 1980s [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. These studies explored the viability of an implicit PIC formulation and its accuracy properties, and resulted in important developments such as the implicit-moment method [6-9, 14, 20] and the direct-implicit method [10,12,15,17,19]. However, limitations of the solver technology at the time forced early implicit PIC practitioners to rely on approximations such as linearization and lagging, which did not respect the strong field-particle coupling.…”

Section: Introductionmentioning

confidence: 99%

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm

Chen

Chacòn

Barnes

2011

Journal of Computational Physics

Self Cite

193

307

View full text Add to dashboard Cite

This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov-Poisson formulation), ours is based on a nonlinearly converged Vlasov-Ampère (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant-Friedrichs-Lewy) and spatial (aliasing) instabilities. It is charge-and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier "energy-conserving" explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton-Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible.

show abstract

“…This realization drove the exploration of implicit PIC starting in the 1980s [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. These studies explored the viability of an implicit PIC formulation and its accuracy properties, and resulted in important developments such as the implicit-moment method [5-8, 13, 20] and the direct-implicit method [9,11,14,17,19]. However, limitations of the solver technology at the time forced early implicit PIC practitioners to rely on approximations such as linearization and lagging, which did not respect the strong field-particle coupling.…”

Section: Introdutionmentioning

confidence: 99%

An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm

Chen

Chacòn

2014

Computer Physics Communications

View full text Add to dashboard Cite

A recent proof-of-principle study proposes a nonlinear electrostatic implicit particle-in-cell (PIC) algorithm in one dimension (Chen, Chacón, Barnes, J. Comput. Phys. 230 (2011) 7018). The algorithm employs a kinetically enslaved Jacobian-free Newton-Krylov (JFNK) method, and conserves energy and charge to numerical round-off. In this study, we generalize the method to electromagnetic simulations in 1D using the Darwin approximation of Maxwell's equations, which avoids radiative aliasing noise issues by ordering out the light wave. An implicit, orbit-averaged time-space-centered finite difference scheme is applied to both the 1D Darwin field equations (in potential form) and the 1D-3V particle orbit equations to produce a discrete system that remains exactly charge-and energy-conserving. Furthermore, enabled by the implicit Darwin equations, exact conservation of the canonical momentum per particle in any ignorable direction is enforced via a suitable scattering rule for the magnetic field. Several 1D numerical experiments demonstrate the accuracy and the conservation properties of the algorithm. IntrodutionThe electromagnetic (EM) Particle-in-cell (PIC) method solves Vlasov-Maxwell's equations for kinetic plasma simulations [1,2]. In the standard approach, Maxwell's equations are solved on a grid, and the Vlasov equation is solved by method of characteristics using a large number of particles, from which the evolution of the probability distribution function (PDF) is obtained. The field-PDF description is tightly coupled. Maxwell's equations (or a subset thereof) are driven by moments of the PDF such as charge density and/or current density. The PDF, on the other hand, follows a hyperbolic equation in phase space, whose characteristics are determined by the fields self-consistently.To date, most PIC methods employ explicit time-stepping (e.g. leapfrog scheme), which can be very inefficient for long-time, large spatial scale simulations. The algorithmic inefficiency of standard explicit PIC is rooted in the presence of numerical stability constraints, which force both a minimum grid-size (due to the so-called finite-grid instability [1,2], which requires resolution of the smallest Debye length) and a very small timestep (due to the wellknown CFL constraint in the general electromagnetic case, c∆t < ∆x, where c is speed of

show abstract

Implicit particle simulation of electromagnetic plasma phenomena

Cited by 5 publications

References 0 publications

A Simplified Implicit Maxwell Solver

A Simplified Implicit Maxwell Solver

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm

An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm

Contact Info

Product

Resources

About