On the Use of Symmetrized Transport Equation in Goal-Oriented Adaptivity

ABSTRACT

In this paper, we revisit the self-adjoint formulation of the transport equation based on the general symmetrization procedure of Marchuk and Agoshkov. In particular, we show how this formulation can be used to obtain solutions of both the forward and the adjoint transport equations with arbitrary source terms from only one solution and one post-processing step. This feature fits well into the well established dual-weighted residual framework for goal-oriented adaptivity, which we use to develop an adaptive finite-element method for solving neutron transport problems. We also describe the relationship to the well known self-adjoint angular flux (SAAF) formulation, allowing us to view the resulting method as an efficient way of performing goal-oriented adaptivity for SAAF. The paper concludes with preliminary numerical experiments that show the viability of the presented method and encourage further research.

KEYWORDS:

• neutron transport equation; self-adjoint formulation; finite element method; Galerkin approximation; goal-oriented adaptivity; symmetrization

Funding

This material is based upon work supported by the Department of Energy, National Nuclear Security Administration, under Award Number(s) DE-NA0002376.

Notes

1 Note that the lack of orthogonality enables us to use |ρ(ϕ_h, ψ_h)| to control the convergence. Using the , in accordance with the analysis of (Jansson et al., 2013), led to similar error estimate, as demonstrated in Section 5.

2 We would like to mention that we did not attempt any optimization of the implementation as this is our first exploration of the method’s capabilities. For instance, the projection problems (17(17) ) are reassembled for every direction, which is clearly unneccessary. Since we use Krylov solvers, another possible optimization could be a matrix-free implementation (as is customary in first-order transport calculations). However, we believe this would not change the conclusions (as it could be applied both to the solution of Equations (16(16) ) and (17(17) )), but only complicate the implementation and limit the choice of preconditioners.