Semi-Implicit Particle-in-Cell Methods Embedding Sparse Grid Reconstructions

In this article, we introduce semi-implicit Particle-In-Cell (PIC) methods based on a discretization of the Vlasov-Maxwell system in the electrostatic regime and embedding sparse grid reconstructions: the Semi-Implict Sparse-PIC (SISPIC) and Energy-Conserving Sparse-PIC (ECSPIC) methods.These schemes are inspired by the Energy-Conserving Semi-Implicit Method (ECSIM) introduced in 39.The particle equations are linearized so that the particle response to the field can be computed by solving a linear system with a stiffness matrix.The two methods feature the three following properties: the scheme is unconditionally stable with respect to the plasma period; the finite grid instability is eliminated, allowing the user to use any desired grid discretization; the statistical error is significantly reduced compared to semi-implicit and explicit schemes with standard grid for the same number of particles.The ECSPIC scheme conserves exactly the discrete total energy of the system but we have experienced numerical instability related to the loss of the field energy non-negativity genuine to the sparse grid combination technique.The SISPIC method is exempted from this instability and is unconditionally stable with respect to the time and spatial discretization, but does not conserve exactly the discrete total energy.The methods have been investigated on a series of two dimensional test cases and gains in term of memory storage and computational time compared to explicit and existing semi-implicit methods have been observed.These gains are expected to be larger for three dimensional computations for which the full potential of sparse grid reconstructions can be achieved.