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

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. (c) 2005 Elsevier B.V. All rights reserved.