Stable Boundary Conditions and Discretization for PN Equations

Abstract

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The number of particles in a finite volume is bounded in time by the number of particles initially occupying the volume augmented by the total number of particles that entered the domain over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic (P N ) approximation, which ensure that this fundamental energy bound is satisfied by the P N approximation. Our BCs are compatible with the characteristic waves of P N equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown based on abstract formulations of P N equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the P N method. We show that summation by parts (SBP) finite differences on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level.

Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering

“…While the current paper focuses on periodic conditions, inflow boundary conditions that prescribe known data on Γ − are more physically relevant. The treatment of such boundary conditions is possible with direct approaches [8] and variational frameworks [15,34]. For vacuum (zero inflow) conditions, an extended computational domain can be employed to maintain sparsity of the P N system [46,16] or to handle complicated geometries [40] in a periodic setting.…”