Energy-conserving Particle-In-Cell scheme based on Galerkin methods with sparse grids

Sparse grid reconstructions have recently been applied to Particle-In-Cell (PIC) methods with a semi-implicit formulation, as demonstrated in [28], to reduce computational costs. By linearizing the particle equations and using a finite difference discretization of the field's equations, along with incorporating sparse grid reconstructions through the combination technique, an exactly energy-conserving scheme was proposed. However, this scheme exhibited numerical instability due to the loss of non-negativity in electric energy, inherent to the combination technique. This paper introduces a novel PIC method with a semi-implicit formulation that embeds sparse grid techniques to exactly conserve discrete total energy, defined as the sum of non-negative kinetic and field energies, ensuring nonlinear stability. The method utilizes a Galerkin approach for the field equations, employing a hierarchical sparse grid representation in the approximation space. This distinguishes it from previous sparse grid PIC methods, which typically use the combination technique and nodal representation. The enhancement of hierarchical subspaces, serving as the truncated combination technique counterpart of the newly introduced method, is proposed to address the limitations of sparse-PIC methods— notably the difficulty in capturing non-smooth and non-axis-aligned solutions. Key features of the method include: unconditional stability with respect to the plasma period; mitigation of grid heating, allowing flexible grid discretization irrespective of the Debye length; exact conservation of discrete total energy; significant reduction in statistical error compared to standard grid schemes for the same number of particles; and decreased computational complexity, particularly in the size of the linear system to be solved. We validate the method through a series of two-dimensional test cases, demonstrating its numerical stability and robust performance.