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

Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 891-924, June 2024.
Abstract. 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-sg) scheme, its standard extension (SISPIC-std), and the energy-conserving sparse-PIC (ECSPIC) scheme. These schemes are inspired by the energy-conserving semi-implicit method introduced in [G. Lapenta, J. Comput. Phys., 334 (2017), pp. 349–366]. 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 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 nonnegativity genuine to the sparse grid combination technique. The SISPIC methods are exempted from this instability and are unconditionally stable with respect to the time and spatial discretization, but do not conserve exactly the discrete total energy. The methods have been investigated on a series of two-dimensional test cases, and gains in terms 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.