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Convergence Error and Higher-Order Sensitivity Estimations

The aim of this study is to improve the accuracy of the finite-difference sensitivities of differential equations solved by iterative methods. New methods are proposed to estimate the convergence error and higher-order sensitivities. The convergence error estimation method is based on the eigenvalue analysis of linear systems, but it can also be used for nonlinear systems. The higher-order sensitivities are calculated by differentiating the approximately constructed differential equation with respect to the design variables. The accuracies of the convergence error and higher-order sensitivity estimation methods are verified using Laplace, Euler, and Navier-Stokes equations. The developed methods are used to improve the accuracy of the finite-difference sensitivity calculations in iteratively solved problems. A bound on the norm value of the finite-difference sensitivity error in the state variables is minimized with respect to the finite-difference step size. The optimum finite-difference step size is formulated as a function of the norm values of both convergence error and higher-order sensitivities. The sensitivities calculated with the analytical and the finite-difference methods are compared. The performance of the proposed methods on the convergence of inverse design optimization is evaluated.