Latest progress on the reduced-order particle-in-cell scheme: I. Refining the underlying formulation

The particle-in-cell (PIC) method is a well-established and widely used kinetic plasma modeling approach that provides a hybrid Lagrangian–Eulerian approach to solve the plasma kinetic equation. Despite its power in capturing details of the underlying physics of plasmas, conventional PIC implementations are associated with a significant computational cost, rendering their applications for real-world plasma science and engineering challenges impractical. The acceleration of the PIC method has thus become a topic of high interest, with several approaches having been pursued to this end. Among these, the concept of reduced-order (RO) PIC simulations, first introduced in 2023, provides a uniquely flexible and computationally efficient framework for kinetic plasma modeling—characteristics that are extensively verified in various plasma configurations. In this two-part article, we report on the latest progress achieved on RO-PIC. Part I revisits the original RO-PIC formulation and introduces refinements that substantially enhance the cost-efficiency and accuracy of the method. We discuss these refinements in comparison against the original formulation, illustrating the progression to a ‘first-order’ implementation from the baseline ‘zeroth-order’ one. In a detailed step-by-step verification, we first test the newly updated reduced-dimension Poisson solver in the first-order RO-PIC against its zeroth-order counterpart using test-case Poisson problems. Next, comparing against the zeroth-order version, we examine the performance of the complete first-order RO-PIC code in two-dimensional plasma problems. One adopted plasma problem corresponds to electron plasma oscillations undergoing Landau damping, and the other to the diocotron instability. The detailed verifications demonstrate that the improvements in the RO-PIC formulation enable the approach to provide full-2D-equivalent results at a substantially lower (up to an order of magnitude) computational cost compared to the zeroth-order RO-PIC.