The Numerov Process Over A Non-Uniform Grid

Abstract The Numerov process is a solution method applicable to some classes of differential equations, that provides an error term of the fifth order in the grid size with a computational cost comparable to that of the finitedifference scheme. In the original formulation of the method, a uniform grid size is required; the paper shows a procedure for extending its applicability to a non-uniform grid in one dimension. The effectiveness of the procedure is tested on a model problem. Next, a variable transformation is described, that reduces the mathematical model of semiconductor devices to a form tractable with the Numerov process. Such a transformation is considered also in the multi-dimensional case, where it is shown that it solves a long-lasting difficulty in semiconductor modeling, namely, the impossibility of reconstructing the current density over the grid elements.