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

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.