Eur. Phys. J. D (2015) 69: 67
https://doi.org/10.1140/epjd/e2015-50226-6

Regular Article

Energy-conserving numerical simulations of electron holes in two-species plasmas^*

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
² Department of Mathematics and Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA

^a e-mail: ycheng@math.msu.edu
^b e-mail: christli@msu.edu
^c e-mail: zhongxh@math.msu.edu

Received: 19 March 2014
Received in final form: 19 December 2014
Published online: 10 March 2015

Abstract

In this paper, we apply our recently developed energy-conserving discontinuous Galerkin (DG) methods for the two-species Vlasov-Ampère system to simulate the evolution of electron holes (EHs). The EH is an important Bernstein-Greene-Kurskal (BGK) state and is constructed based on the Schamel distribution in our simulation.Even though the knowledge of steady state EHs has advanced significantly, little is known about the full dynamics of EHs that nonlinearly interact with ions in plasmas. In this paper, we simulate the full dynamics of EHs with DG finite element methods, coupled with explicit and implicit time integrators. Our methods are demonstrated to be conservative in the total energy and particle numbers for both species. By varying the mass and temperature ratios, we observe the stationary and moving EHs, as well as the break up of EHs at later times upon initial perturbation of the electron distribution. In addition, we perform a detailed numerical study for the BGK states for the nonlinear evolutions of EH simulations. Our simulation results should help to understand the dynamics of large amplitude EHs that nonlinearly interact with ions in space and laboratory plasmas.