Comparison of semi-implicit and explicit particle in cell methods for the study of turbulence in space plasmas
Authors: Pucci Francesco , Parashar Tulasi N. , Matthaeus William H. , Lapenta Giovanni 
Published: March 2020

&lt;p&gt;The plasma that permeates the solar wind, the solar corona, the Earth's magnetosheath and several&lt;br&gt;other space environments is in a turbulent state. The effect of turbulence on the dynamics of such systems&amp;#160;&lt;br&gt;is very relevant, considering that it is invoked to explain plasma heating, and particle acceleration and transport&lt;br&gt;in those environments.&lt;br&gt;From a mathematical point of view, turbulence is a non linear phenomenon whose study, in the kinetic&amp;#160;&lt;br&gt;description of plasmas, requires the solution of the non linear Vlasov-Maxwell system of equations.&amp;#160;&lt;br&gt;Due to the complexity of the problem, the solutions are nowadays found mainly by means of numerical simulations.&amp;#160;&lt;br&gt;The most widely used method for the solution of the Vlasov-Maxwell system is the Particle In Cell (PIC) method. &amp;#160;&amp;#160;&lt;br&gt;PIC methods can be divided into two major classes: explicit and implicit, depending on the algorithm used&lt;br&gt;for advancing the solution in time.&lt;br&gt;In this work, we compare two different PIC methods that use an explicit and a semi-implicit algorithm, respectively.&amp;#160;&lt;br&gt;The explicit method is implemented in the code P3D[1], while the semi-implicit method in code iPic3D[2].&lt;br&gt;Both methods are fully kinetic, namely they retain the kinetic effects for both ions and electrons.&amp;#160;&lt;br&gt;The two codes are tested against a classical set up of plasma turbulence in a 2D cartesian&amp;#160;&lt;br&gt;geometry[3]. The system is initialized with a restricted number of modes at large scale and&lt;br&gt;evolves in time without forcing. The box size is of several tens of ion inertial length. The&amp;#160;&lt;br&gt;grid size is of the order of the Debye length for the explicit scheme, to ensure numerical stability,&lt;br&gt;and is varied across the electron skin depth for the semi-implicit, by performing different simulations.&lt;br&gt;Several analyses are presented: global energy conversion, magnetic and electric spectra, scale dependent kurtosis,&amp;#160;&lt;br&gt;temperature anisotropy for both species, proxies of dissipation such as J.E and PiD[4].&lt;br&gt;The weaknesses and strengths of the two methods in terms of description of the physical dynamics and of&amp;#160;&lt;br&gt;computational time are presented, along with a convergence study of the semi-implicit to the explicit&lt;br&gt;method as the resolution of the former is varied.&amp;#160;&lt;/p&gt;&lt;p&gt;[1] Zeiler, A., Biskamp, D., Drake, J. F., Rogers, B. N., Shay, M. A., &amp; Scholer, M. (2002). Journal of Geophysical Research: Space Physics, 107(A9), SMP-6.&lt;br&gt;[2] Markidis, S., Lapenta, G., &amp; Rizwan-uddin (2010). Mathematics and Computers in Simulation, 80(7), 1509-1519.&lt;br&gt;[3] Parashar, T. N., Matthaeus, W. H., &amp; Shay, M. A. (2018). The Astrophysical Journal Letters, 864(1), L21.&lt;br&gt;[4] Yang, Y., Matthaeus, W. H., Parashar, T. N., Wu, P., Wan, M., Shi, Y., et al. (2017). Physical Review E, 95(6), 061201.&lt;/p&gt;