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

Plasma Physics;Instrumentation and Methods for Astrophysics