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

TL;DR

This paper investigates how inaccuracies in the RHS of the Poisson equation, caused by charge sampling errors near boundaries in PIC simulations, affect numerical solutions and compares linear and quadratic schemes.

Contribution

It provides analytical and numerical analysis of RHS inaccuracies' effects on discretization errors and proposes a calibration method to improve quadratic scheme accuracy.

Findings

Linear scheme performs better under RHS inaccuracies.

RHS inaccuracies alter local truncation errors differently for linear and quadratic schemes.

A calibration strategy can restore quadratic scheme accuracy.

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.