Applying second-order adjoint perturbation theory to time-dependent problems

Abstract

In this paper we discuss the application of second-order adjoint based perturbation techniques to linear and nonlinear time dependent problems. These techniques are based on the property of the adjoint problem which allows calculating the set of first and second order coefficients by solving a number of adjoint systems.The derivation of a second order adjoint theory that can be used to calculate the second order Taylor components of a functional response is first presented for nonlinear problems, and then reformulated for linear systems. This implementation is based on the Adjoint Sensitivity Analysis Procedure, originally presented by Cacuci.The theoretical part is followed by the application of the technique to some classical reactor physics problems: a fuel depletion calculation, a linear and a nonlinear point-kinetic problem.The results obtained using the second order theory are compared with the values obtained using a first order approximation and a direct solution of the forward problem. Our first results show that the procedure provides good estimations in presence of higher order perturbation components, being able to reconstruct the responses of interest even in presence of large perturbations. The main differences between linear and nonlinear formulations and their computational implications are also discussed.