Cartesian Coordinate, Oblique Boundary, Finite Differences and Interpolation

TL;DR

This paper presents a second-order accurate numerical scheme for solving Poisson's equation on 3D Cartesian grids with oblique boundaries, including boundary conditions and dielectric interfaces, suitable for particle-in-cell simulations.

Contribution

It introduces a compact difference scheme that accurately handles oblique boundaries and media interfaces with detailed implementation for Robin conditions.

Findings

Scheme achieves second-order accuracy in potential and field interpolation.

Numerical tests confirm the scheme's accuracy and proper scaling.

Effective for particle-in-cell codes requiring precise force calculations.

Abstract

A numerical scheme is described for accurately accommodating oblique, non-aligned, boundaries, on a three-dimensional cartesian grid. The scheme gives second-order accuracy in the solution for potential of Poisson's equation using compact difference stencils involving only nearest neighbors. Implementation for general "Robin" boundary conditions and for boundaries between media of different dielectric constant for arbitrary-shaped regions is described in detail. The scheme also provides for the interpolation of field (potential gradient) which, despite first-order peak errors immediately adjacent to the boundaries, has overall second order accuracy, and thus provides with good accuracy what is required in particle-in-cell codes: the force. Numerical tests on the implementation confirm the scalings and the accuracy.