Convergence Error and Higher-Order Sensitivity Estimations

Eyi S.

AIAA JOURNAL, vol.50, no.10, pp.2219-2234, 2012 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 10
  • Publication Date: 2012
  • Doi Number: 10.2514/i.j051592
  • Journal Name: AIAA JOURNAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2219-2234
  • Middle East Technical University Affiliated: Yes


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.