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

The energy-conserving semi-implicit (ECsim) method presented by the author 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 linearly express 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. Here, three methods that modify the construction of the mass matrix are analyzed. First, the paper considers how the sub-cycling of the particle motion modifies the mass matrix. Second, a form of smoothing that reduces the noise while retaining exact energy conservation is introduced. Finally, an approximation of the mass matrix is discussed that transforms the ECsim scheme to the implicit moment method.