The solution of nonlinear boundary-value transport problems in electron and plasma devices

Abstract

A method of solution of the nonlinear Landau-Vlasov equation which occurs in charged particle transport phenomena is described. The method is similar in concept to the Spherical Harmonics Method of neutron transport theory, since the particle distribution function is expanded in orthogonal functions of velocity to obtain an infinite set of partial differential equations. The nonlinearity arises when the particle interactions are taken into account in the force term of the Landau-Vlasov equation. The latter is introduced into the equation iteratively and a numerical integration routine is described to integrate the set of equations resulting from a finite ( nth order) truncation of the expansion. The truncated equations, which are for the first n velocity moments of the distribution function, are such that these moments satisfy the untruncated equations also, whereas higher moment equations are modified by a fictitious source term. The convergence of the finite-difference equations and the convergence of the finite-difference solution to the solution of the differential equations is proved for the linear case in which the force is assumed to be known. Computational evidence of the convergence in the nonlinear case is provided, although no mathematical proof has so far been developed.