Advances in the Implementation of the Exactly Energy Conserving Semi-Implicit (ECsim) Particle-in-Cell Method

The Energy Conserving semi-implicit method (ECsim), presented by Lapenta in 2017, is a Particle Cell (PIC) algorithm for the simulation of plasmas. conservation achieved within formulation that does not require any non-linear solver. A mass matrix introduced to express linearly particle-field coupling. With preserves energy machine precision. construction central nature and also main cost computational cycle. We analyze here three methods modify matrix. First, we consider how sub-cycling particle motion modifies Second, introduce form smoothing reduces noise while retaining exact conservation. Finally, discuss an approximation transform ECsim scheme implicit moment method.