Numerical Error Analysis of the Poisson Equation under RHS Inaccuracies in Particle-in-Cell Simulations

TL;DR

This work analyzes how right-hand-side (RHS) inaccuracies, typical of charge-density sampling near irregular Dirichlet boundaries in PIC simulations, affect the Poisson solver on Cartesian meshes with embedded boundaries. Through 1D explicit error derivations and 2D truncation-error analyses, it shows RHS underestimation can reduce linear boundary truncation while introducing zeroth-order terms in the quadratic scheme, making the linear method often more accurate in practice. Numerical experiments in 1D, 2D, and 3D confirm these trends and demonstrate that a simple RHS calibration based on local averages, $\overline{\delta}$, can restore the quadratic scheme’s accuracy. The findings offer practical guidance for selecting boundary treatments in PIC contexts and provide insight into how boundary-induced RHS errors interact with discretization accuracy in Poisson-type problems.

Abstract

Particle-in-Cell (PIC) simulations rely on accurate solutions of the electrostatic Poisson equation, yet accuracy often deteriorates near irregular Dirichlet boundaries on Cartesian meshes. While much research has addressed discretization errors on the left-hand side (LHS) of the Poisson equation, the impact of right-hand-side (RHS) inaccuracies - arising from charge density sampling near boundaries in PIC methods - remains largely unexplored. This study analyzes the numerical errors induced by underestimated RHS values at near-boundary nodes when solving the Poisson equation using embedded boundary finite difference schemes with linear and quadratic treatments. Analytical derivations in one dimension and truncation error analyses in two dimensions reveal that such RHS inaccuracies modify local truncation behavior differently: they reduce the dominant truncation error in the linear scheme but introduce a zeroth-order term in the quadratic scheme, leading to larger global errors. Numerical experiments in one-, two-, and three-dimensional domains confirm these findings. Contrary to expectations, the linear scheme yields superior overall accuracy under typical PIC-induced RHS inaccuracies. A simple RHS calibration strategy is further proposed to restore the accuracy of the quadratic scheme. These results offer new insight into the interplay between boundary-induced RHS errors and discretization accuracy in Poisson-type problems.

Figures (21)

• Figure 1: 1D problem illustration with a Dirichlet boundary positioned between nodes.
• Figure 2: RHS inaccuracy encountered in PIC simulations, illustrated in 1D
• Figure 3: 2D illustration for RHS value treatment for a near-boundary node $(i,j)$
• Figure 4: comparison of the leading error terms in $\tau_1$ among four scenarios.
• Figure 5: comparison of the leading error terms in $\xi_i^L$ among the four scenarios.