Discretization of Macroscopic Transport Equations on Non-Cartesian Coordinate Systems

We discuss discretization schemes for the Poisson equation, the isothermal drift-diffusion equations, and higher order moment equations derived from the Boltzmann transport equation for general coordinate systems. We briefly summarize the method of dimension reduction when the problem does not depend on one coordinate. Discretization schemes for dimension-reduced coordinate systems are introduced, which provide curvilinear coordinate systems. In addition to the reduction of the dimensionality, another benefit of these curved coordinate systems is that the domain approximation is more accurate, and therefore, the mesh point density can be kept smaller compared to the original problem. We obtain a discretization scheme for the isothermal drift-diffusion equation in closed from. For higher order transport equations, we use the approximation method of optimum artificial diffusivity and generalize it for non-Cartesian coordinate systems. For the special case of cylindrical coordinates, we can show that it is not necessary to introduce special discretization schemes apart from the standard Scharfetter-Gummel scheme.