Streamline upwind Petrov–Galerkin methods for the steady-state Boltzmann transport equation

Abstract

This paper describes Petrov–Galerkin finite element methods for solving the steady-state Boltzmann transport equation. The methods in their most general form are non-linear and therefore capable of accurately resolving sharp gradients in the solution field. In contrast to previously developed non-linear Petrov–Galerkin (shock-capturing) schemes the methods described in this paper are streamline based. The methods apply a finite element treatment for the internal domain and a Riemann approach on the boundary of the domain. This significantly simplifies the numerical application of the scheme by circumventing the evaluation of complex half-range angular integrals at the boundaries of the domain. The underlying linear Petrov–Galerkin scheme is compared with other Petrov–Galerkin methods found in the radiation transport literature such as those based on the self-adjoint angular flux equation (SAAF) [G.C. Pomraning, Approximate methods of solution of the monoenergetic Boltzmann equation, Ph.D. Thesis, MIT, 1962] and the even-parity (EP) equation [V.S. Vladmirov, Mathematical methods in the one velocity theory of particle transport, Atomic Energy of Canada Limited, Ontario, 1963]. The relationship of the linear method to Riemann methods is also explored.The Petrov–Galerkin methods developed in this paper are applied to a variety of 2-D steady-state and fixed source radiation transport problems. The examples are chosen to cover the range of different radiation regimes from optically thick (diffusive and/or highly absorbing) to transparent and semi-transparent media. These numerical examples show that the non-linear Petrov–Galerkin method is capable of producing accurate, oscillation free solutions across the full spectrum of radiation regimes.