Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow

Abstract An error estimation and grid adaptive strategy is presented for estimating and reducing simulation errors in functional outputs of partial differential equations. The procedure is based on an adjoint formulation in which the estimated error in the functional can be directly related to the local residual errors of both the primal and adjoint solutions. This relationship allows local error contributions to be used as indicators in a grid-adaptive strategy designed to produce specially tuned grids for accurately estimating the chosen functional. In this paper, attention is limited to one-dimensional problems, although the procedure is readily extendable to multiple dimensions. The error estimation procedure is applied to a standard, second-order, finite volume discretization of the quasi-one-dimensional Euler equations. Both isentropic and shocked flows are considered. The chosen functional of interest is the integrated pressure along a variable-area duct. The error estimation procedure, applied on uniform grids, provides superconvergent values of the corrected functional. Results demonstrate that additional improvements in the accuracy of the functional can be achieved by applying the proposed adaptive strategy to an initially uniform grid. The proposed adaptive strategy is also compared with a standard adaptive scheme based on the interpolation error in the computed pressure. The proposed scheme consistently yields more accurate functional predictions than does the standard scheme.