A Self-Adjoint Variational Principle for Deriving Vacuum and Interface Boundary Conditions in the Spherical Harmonics Method

By the technique of splitting the total directional flux into even and odd portions in angle, the stationary monoenergetic Boltzmann equation with arbitrary collision kernel and with arbitrary external directional source of a general geometry is symmetrized to a self-adjoint form. The continuity and boundary conditions for the resulting self-adjoint integro-differential equation are explicitly constructed. A variational principle is then set up by devising a self-adjoint Lagrangian whose minimum property is equivalent to the symmetrized Boltzmann equation with the associated continuity and boundary conditions. The developed variational principle contains no arbitrariness and is used for deriving unique variational boundary conditions for the P₁ approximation of the spherical harmonics method. It is shown, for a general geometry, that applying the semidirect variational method with an angle-independent trial function yields, without any physical reasoning, the correct P₁ differential equation and the corresponding no-return-current boundary condition.