Work

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

Year: 2026

Type: article

Abstract: Particle-in-Cell (PIC) simulations require accurate solutions of the electrostatic Poisson equation, yet accuracy often degrades near irregular Dirichlet boundaries on Cartesian meshes. While prior work has focused on left-hand-side (LHS) discretization errors, the impact of right-hand-side (RHS) inaccuracies arising from charge deposition near boundaries remains largely unexplored. This study analyzes numerical errors induced by underestimated RHS values at near-boundary nodes when using embedded finite difference schemes with linear and quadratic boundary treatments. Analytical results in one dimension and truncation error analyses in two dimensions show that RHS inaccuracies affect the two schemes in fundamentally different ways: They reduce boundary-induced errors in the linear scheme but introduce zeroth-order truncation errors in the quadratic scheme, leading to larger global errors. Numerical experiments in one, two, and three dimensions confirm these predictions. In two-dimensional tests, RHS inaccuracies reduce the L∞ error of the linear scheme by a factor of 2–3, while increasing the quadratic-scheme error by several times, and in some cases by nearly an order of magnitude, with both schemes retaining second-order global convergence. A simple δ¯-based RHS calibration is proposed and shown to effectively restore the accuracy of the quadratic scheme.

Source: Computation

DOI: 10.3390/computation14010013

Authors: Kai Zhang, Tao Xiao, Weizong Wang, Bijiao He

Institutions: Hunan University of Arts and Science, Beihang University

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,172

Funders: National Natural Science Foundation of China, Natural Science Foundation of Hunan Province