A self-adjoint form of the linear transport equation

Abstract

A variety of approximation procedures are based upon self-adjoint principles in the solution of linear boundary-value problems [ 11. If these methods are applied to the transport operator, the original equation has to be transformed to an equivalent self-adjoint form. Such a transformation was first performed by Vladimirov in [2]. More recent numerical work along these lines has been reported in connection with the conventional variational principles [3] and the finite-element method [4, 51. The one-group transformations in [2]-[6] were concerned with an operator consisting of an antisymmetric gradient term. In this paper we wish to consider the general energy-dependent case with a nonsymmetric integral kernel that additionally contributes to the non-self-adjoint nature of the problem. In the subsequent sections sufficient conditions are derived that ensure the existence of the self-adjoint transport operator analogous to one constructed in [6] for the one-speed case. This operator is positive definite which facilitates the use of standard variational-approximation techniques ([l], [4], [6]). While no numerical work is presented we attempt to demonstrate the utility of the formalism by discretizing the limiting conditions. From given multigroup data it can be readily verified whether these conditions are satisfied. In view of [4] t i is interesting to note that the source-iteration scheme imposes less stringent limitations on the physical parameters than the rigorous conditions obtained here.