Numerical solution of the cylindrical Poisson equation using the Local Taylor Polynomial technique

Simulation of charged-particle beam devices using particle-in-cell methods in (r, z) cylindrical coordinates can be extremely efficient, incorporating a majority of the physics and yielding high resolution with minimal computational costs. However, accuracy requires adjustment of solution coefficients and particle/current weights with radius, especially near the cylindrical axis. Prior algorithms have been based on special models and approximations with no consistent method for computing coefficients in general cases. The Local Taylor Polynomial (LTP) technique presented in this paper provides a general framework for determination of coefficients and development of PDE solution algorithms. The LTP method employs a local (polynomial) solution of the continuum PDE to construct numerical algorithms. As a result, LTP systematically generates accurate coefficients and incorporates fields and sources on an equal basis. This paper focuses on the scalar Poisson PDE in cylindrical coordinates to illustrate the LTP technique, coefficient derivation, electric field calculation and handling of space charge effects. Application of LTP to non-conformal boundaries and non-uniform grids is presented. Although focused on cylindrical coordinates in this paper, the LTP technique has general applicability to other PDE’s and geometries, and is broadly applicable to problems in beam/field simulation.