Adjoint eigenvalue correction for elliptic and hyperbolic neutron transport problems

Abstract

An adjoint-based a posteriori error measure is developed and applied to the Keff eigenvalue in particle transport problems using the diffusion approximation and full transport solutions. This demonstrates application of an eigenvalue error recovery scheme that can be applied to both elliptic and hyperbolic operators. The Keff eigenvalue is first obtained via a conventional inverse power iteration on the fission source, from the forward system of equations using a simple linear finite element type. The solution procedure is then repeated using the adjoint equations. The eigenvector solution to the adjoint system is enriched in a post-processor step, and convolved with the residual of the forward equations. This produces a computable approximation to the error in the eigenvalue. This approximation to the error is then subtracted from the eigenvalue producing a better estimate. It is shown how this approach can accelerate the mesh convergence of the eigenvalue in both smooth, diffusive problems using an elliptic operator and also in non-smooth transport problems in which the operator is of hyperbolic form. In the elliptic case, the diffusion equation is discretised with continuous finite elements. In the hyperbolic case, the Boltzmann Transport Equation is discretised with discontinuous Galerkin weighted finite elements. The approach to recovering the error in the Keff eigenvalue is common to both cases.