Gauss's Law satisfying Energy-Conserving Semi-Implicit Particle-in-Cell method

Abstract The Energy Conserving Semi-Implicit Method (ECSIM) introduced by Lapenta (2017) has many advantageous properties compared to the classical semi-implicit and explicit PIC methods. Most importantly, energy conservation eliminates the growth of the finite grid instability. We have implemented ECSIM in a different and more efficient manner than the original approach. More importantly, we have addressed two major shortcomings of the original ECSIM algorithm: there is no mechanism to enforce Gauss's law and there is no mechanism to reduce the numerical oscillations of the electric field. A classical approach to satisfy Gauss's law is to modify the electric field and its divergence using either an elliptic or a parabolic/hyperbolic correction based on the Generalized Lagrange Multiplier method. This correction, however, violates the energy conservation property, and the oscillations related to the finite grid instability reappear in the modified ECSIM scheme. We invented a new alternative approach: the particle positions are modified instead of the electric field in the correction step. Displacing the particles slightly does not change the energy conservation property, while it can satisfy Gauss's law by changing the charge density. We found that the new Gauss's Law satisfying Energy Conserving Semi-Implicit Method (GL-ECSIM) produces superior results compared to the original ECSIM algorithm. In some simulations, however, there are still some numerical oscillations present in the electric field. We attribute this to the simple finite difference discretization of the energy conserving implicit electric field solver. We modified the spatial discretization of the field solver to reduce these oscillations while only slightly violating the energy conservation properties. We demonstrate the improved quality of the GL-ECSIM method with several tests.

[1]  E. S. Weibel,et al.  Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution , 1959 .

[2]  Stefano Markidis,et al.  Scaling the Ion Inertial Length and Its Implications for Modeling Reconnection in Global Simulations , 2017 .

[3]  Paolo Ricci,et al.  A simplified implicit maxwell solver , 2002 .

[4]  Liang Wang,et al.  Comparison of multi-fluid moment models with Particle-in-Cell simulations of collisionless magnetic reconnection , 2014, 1409.0262.

[5]  R. W. Hockney,et al.  Body-fitted electromagnetic PIC software for use on parallel computers , 1995 .

[6]  Hiroshi Matsumoto,et al.  A new charge conservation method in electromagnetic particle-in-cell simulations , 2003 .

[7]  Stefano Markidis,et al.  Two-way coupling of a global Hall magnetohydrodynamics model with a local implicit particle-in-cell model , 2014, J. Comput. Phys..

[8]  Motohiko Tanaka Macroscale implicit electromagnetic particle simulation of magnetized plasmas , 1988 .

[9]  Tamas I. Gombosi,et al.  A six-moment multi-fluid plasma model , 2019, J. Comput. Phys..

[10]  C. Birdsall,et al.  Plasma Physics via Computer Simulation , 2018 .

[11]  Luis Chacón,et al.  A curvilinear, fully implicit, conservative electromagnetic PIC algorithm in multiple dimensions , 2016, J. Comput. Phys..

[12]  Claus-Dieter Munz,et al.  Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model , 2000 .

[13]  Pierre Degond,et al.  On a finite-element method for solving the three-dimensional Maxwell equations , 1993 .

[14]  D. A. Knoll,et al.  A 2D high-ß Hall MHD implicit nonlinear solver , 2003 .

[15]  B. M. Marder,et al.  A method for incorporating Gauss' lasw into electromagnetic pic codes , 1987 .

[16]  A. Bruce Langdon,et al.  On enforcing Gauss' law in electromagnetic particle-in-cell codes , 1992 .

[17]  Stefano Markidis,et al.  The energy conserving particle-in-cell method , 2011, J. Comput. Phys..

[18]  J. W. Eastwood,et al.  The virtual particle electromagnetic particle-mesh method , 1991 .

[19]  William Daughton,et al.  The inversion layer of electric fields and electron phase-space-hole structure during two-dimensional collisionless magnetic reconnection , 2011 .

[20]  S. Markidis,et al.  Global Three‐Dimensional Simulation of Earth's Dayside Reconnection Using a Two‐Way Coupled Magnetohydrodynamics With Embedded Particle‐in‐Cell Model: Initial Results , 2017, 1704.03803.

[21]  Luis Chacón,et al.  A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell algorithm , 2015, Comput. Phys. Commun..

[22]  Motohiko Tanaka The macro-EM particle simulation method and a study of collisionless magnetic reconnection , 1994 .

[23]  Michael Hesse,et al.  Geospace Environmental Modeling (GEM) magnetic reconnection challenge , 2001 .

[24]  S. Markidis,et al.  Extended magnetohydrodynamics with embedded particle‐in‐cell simulation of Ganymede's magnetosphere , 2014 .

[25]  John D. Villasenor,et al.  Rigorous charge conservation for local electromagnetic field solvers , 1992 .

[26]  Igor V. Sokolov,et al.  Alternating-order interpolation in a charge-conserving scheme for particle-in-cell simulations , 2011, Comput. Phys. Commun..

[27]  Zuyin Pu,et al.  Current structure and flow pattern on the electron separatrix in reconnection region , 2017, Geoscience Letters.

[28]  T. Esirkepov,et al.  Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor , 2001 .

[29]  Stefano Markidis,et al.  Multi-scale simulations of plasma with iPIC3D , 2010, Math. Comput. Simul..

[30]  Luis Chacón,et al.  An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm , 2011, J. Comput. Phys..

[31]  William Daughton,et al.  Phase space structure of the electron diffusion region in reconnection with weak guide fields , 2012 .

[32]  R. Morse,et al.  NUMERICAL SIMULATION OF THE WEIBEL INSTABILITY IN ONE AND TWO DIMENSIONS. , 1971 .

[33]  G. Lapenta,et al.  Multiple-scale kinetic simulations with the energy conserving semi-implicit particle in cell method , 2016, 1612.08289.

[34]  Diego Gonzalez-Herrero,et al.  Performance analysis and implementation details of the Energy Conserving Semi-Implicit Method code (ECsim) , 2018, Comput. Phys. Commun..

[35]  Giovanni Lapenta,et al.  Exactly energy conserving semi-implicit particle in cell formulation , 2016, J. Comput. Phys..