Nonlinear perturbative electromagnetic (Darwin) particle simulation of high intensity beams

Computer Physics Communications

2015

For decades, the Vlasov-Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. Here, we explore a fully implicit PIC algorithm for the Vlasov-Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space-time-centered, employing particle sub-cycling and orbit-averaging. The algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large time steps and cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. We demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D-3V.

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Section: Electromagnetic Vlasov-darwin Modelmentioning

confidence: 99%

“…For exact charge conservation, we combine the charge-current interpolation scheme (Eqs. [12][13][14]38) employed in this study with the cell-crossing scheme introduced by Ref. [49].…”

Section: = [ρ]mentioning

confidence: 99%

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A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm

Computer Physics Communications

2015

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“…Nielson and Lewis [28] introduced semiimplicit schemes to advance the Darwin-PIC system, which have become the standard for later development and applications of plasma Darwin-PIC simulations (see Refs. [21,[29][30][31][32][33][34][35][36] and references therein). Nevertheless, the resulting field equations, in either Hamiltonian or Lagrangian form, are complicated and difficult to solve, especially when non-periodic boundary conditions are employed [31,32,37,38], and feature no exact conservation properties (e.g., local charge, total energy, or total momentum).…”

Section: Introdutionmentioning

confidence: 99%

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

Computer Physics Communications

2014

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

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Energy-conserving perfect-conductor boundary conditions for an implicit, curvilinear Darwin particle-in-cell algorithm

Journal of Computational Physics

2019