Generalized Ohm's law and potential equation in computational weakly-ionized plasmadynamics

A variant of the generalized Ohm's law that is suited for a weakly-ionized multicomponent plasma in a magnetic field is here derived. The latter takes into consideration the current due to the non-neutrality of the plasma, the current due to the Hall effect, and the currents due to the ion slip associated with each type of ion. An equation for the electric field potential applicable to a non-neutral multicomponent plasma in the presence of a magnetic field is then presented. Despite some similarities between the potential equation and the Poisson equation, it is argued that the discretization of the potential equation cannot be accomplished in the same manner by using only central differences. It is here proven (and subsequently verified through a test case) that when the plasma exhibits conjunctly a high Hall parameter and a high electrical conductivity gradient, the centered stencils introduce spurious oscillations which can lead to severe numerical error. A novel discretization of the potential equation consisting of a blend of central and upwind differences is then presented. The proposed scheme is consistently monotonic for any value of the Hall parameter and is second-order accurate except in the vicinity of discontinuities.