The Semiconductor Model Solved by the Numerov Process over a Non-Uniform Grid

The Numerov Process (NP) provides the solution of some classes of ODEs with an accuracy much superior to that of the standard finite-difference or box-integration methods. The original formulation of NP requires a uniform grid, which is a drawback for applications to, e.g., the semiconductor-device equations. Purpose of this work is showing how a method for extending NP to a non-uniform grid is applied to the solution of the drift-diffusion model. The method keeps the fifth-order accuracy of the original NP. In the multi-dimensional case, the variable transformation illustrated in the paper is found beneficial also when standard solution schemes are used; in fact, it makes the current-density vector well defined within each grid element.