A curvilinear, fully implicit, conservative electromagnetic PIC algorithm in multiple dimensions

Abstract

We extend a recently proposed fully implicit PIC algorithm for the Vlasov-Darwin model in multiple dimensions (Chen and Chacón (2015) [1]) to curvilinear geometry. As in the Cartesian case, the approach is based on a potential formulation (φ, A), and overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. Conservation theorems for local charge and global energy are derived in curvilinear representation, and then enforced discretely by a careful choice of the discretization of field and particle equations. Additionally, the algorithm conserves canonical-momentum in any ignorable direction, and preserves the Coulomb gauge ∇ · A = 0 exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large 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 numerical experiments in mapped meshes in 1D-3V and 2D-3V.

A fully implicit, conservative, non-linear, electromagnetic hybrid particle-ion/fluid-electron algorithm

“…One of the key successes of implicit particle methods for the full kinetic PIC model is the capability for discrete energy conservation using a finite time-step [22,23] (the earlier explicit method of Ref.[24] conserved energy only in the limit of ∆t → 0). It has been demonstrated that discrete energy conservation can be found in addition to local charge conservation in multi-dimensional [25] curvilinear geometries [26] with sub-cycling and orbit averaging. Recently, there has been particular interest in discrete methods that preserve geometric structures of the Vlasov-Maxwell system [27,28,29,30,31].An example of the macroscopic consequences of violating discrete conservation in fully kinetic PIC algorithms is the finite-grid instability, caused by aliasing errors between the particles, which live in continuous space, and the discrete spatial grid.…”

A particle-in-cell method for the simulation of plasmas based on an unconditionally stable field solver

“…This further motivates the development of higher-order accurate versions of our method, which markedly improve the wave propagation characteristics of the implicit wave solver and would allow for the use of a larger time step size [5]. By drawing from the rich literature on the issues confronting existing PIC simulation methods (including especially the works reviewed in the introduction), the authors believe that the method presented in this paper could provide a framework to address the development of stable schemes that are higher-order accurate in time and space, the accurate treatment of complex boundary geometries, the implementation of grid refinement and domain decomposition schemes to enable efficient parallel computation, and the removal of time and grid step size restrictions through the development of implicit particle models and hybrid kinetic-fluid PIC simulation techniques appropriate to specific appliations [66,67,17,68,33,35,69,70,71,72,73,74,75,76,77,78,79,38,80], in order to advance plasma simulation capabilities beyond their present state.…”

Multiscale high-order/low-order (HOLO) algorithms and applications

“…Exact energy conservation can be achieved discretely when an exactly time-centered (Crank-Nicolson, CN) temporal discretization scheme is employed for both particles (per subcycle) and fields [61,81,19,20]. All those conservation properties can be naturally generalized to curvilinear geometries [82,83].…”