Reduction of discretisation errors in the numerical simulation of axisymmetric electrostatic problems

Abstract

Abstract The calculation of electrostatic fields by a difference approximation of the partial derivatives in the nodes of a superimposed grid is a well established method. With more memory available on modern computers, the discretisation can be refined by using more and more mesh points. This raises the question, to what extent the discretisation error caused by a difference approximation accumulates on large grids and how to reduce it. In this paper, we investigate the absolute error in a field calculation for the example of a spherical condensor, using single and double precision arithmetics and various difference approximations. Especially in the calculation of axisymmetric problems, there is quite often a need for more resolution of the mesh in the vicinity of the symmetry axis (field emission guns, thin beams with space charge). A nonconformal mapping by a logarithmic transformation of the radial coordinate allows to superimpose a calculation grid over an arbitrary span of magnitudes and also improves the discretisation error, because the linear derivative of the radial coordinate in the Laplace operator is eliminated. Additionally, a Moebius transformation is provided to study open boundary problems and with the simple solver (SOR) used, dielectric boundaries and narrow gaps are handled with ease and in complete generality by the use of the boundary processor POLYGON [5].