Continuity Conditions for a Homogeneous or Heterogeneous P_N Approximation

The problem of obtaining continuity conditions for a P_N approximation is approached from a variational point of view. A functional is defined that admits piecewise discontinuous trial functions and has the transport equation and flux continuity conditions as its Euler equations. A reduced functional, formed by adopting a truncated spherical-harmonics expansion as a trial function, has as its Euler equations the P_N equations and approximate flux-continuity conditions. These variational continuity conditions, which involve full-range angular integrals, are seen to be the same as those of Rumyantsev. Marshak continuity conditions, which involve half-range angular integrals obtained by Marshak matching, are shown to be equivalent to Rumyantsev's continuity conditions. Continuity conditions for a heterogeneous P_N approximation are obtained by extending the notion of Marshak matching are shown to the case where a P_N,approximation is employed in an adjacent region.