A projection method for particle resampling

TL;DR

The paper tackles particle noise and grid-quality issues in high-dimensional kinetic simulations by introducing a projection-based resampling method that maps between particle representations and finite element spaces while exactly preserving moments up to the FE space degree. The approach uses a Moore–Penrose pseudoinverse to compute new particle weights after resampling, ensuring field quantities remain unchanged and enabling remapping to diverse particle layouts. Through PETSc-PIC-based experiments on 1X+1V Landau damping and two-stream instabilities, the method demonstrates noise suppression, improved long-time stability, and preserved phase-space structure, with direct remap and pseudoinverse variants illustrating the benefits and limitations. The work also outlines practical directions for future adaptivity, entropy control, AMR integration, and applications to more complex plasma models, highlighting potential for scalable, structure-preserving PIC–continuum coupling in high-dimensional systems.

Abstract

Particle discretizations of partial differential equations are advantageous for high-dimensional kinetic models in phase space due to their better scalability than continuum approaches with respect to dimension. Complex processes collectively referred to as particle noise hamper long time simulations with particle methods. One approach to address this problem is particle mesh adaptivity or remapping, known as particle resampling. This paper introduces a resampling method that projects particles to and from a (finite element) function space. The method is simple; using standard sparse linear algebra and finite element techniques, it can adapt to almost any set of new particle locations and preserves all moments up to the order of polynomial represented exactly by the continuum function space. This work is motivated by the Vlasov-Maxwell-Landau model of magnetized plasmas with up to six dimensions, 3X in physical space and 3V in velocity space, and is developed in the context of a 1X + 1V Vlasov-Poisson model of Landau damping with logically regular particle and continuum phase space grids. Stable long time dynamics are demonstrated up to T = 500 and reproducibility artifacts and data with stable dynamics up to T = 1000 are publicly available.

Figures (12)

• Figure 1: E field on $64 \times 128$ particle grid (y-axis is velocity): uniform distribution (left); r-refinement (right)
• Figure 1: Field amplitude with linear Landau damping, $A=0.01$: no resampling (blue and noisy), a direct remapping finite element version of Myers et al. (magenta), and the projection method (red)
• Figure 1: $E_{max}$, Two-stream instability resampling period comparison test.
• Figure 1: $E_{max}, A = 0.0001$, Refinement in particles per cell long time simulations
• Figure 1: $E_{max}, A = 0.0001$, resampling period "anti-convergence" study (left), instability detail (right)