Numerical and theoretical considerations for sensitivity calculation of discontinuous flow,.pdf

Numerical and theoretical considerations for sensitivity calculation of discontinuous flow Chris Homescu, I.M. Navon ? Departmentt of Mathematics and School of Computational Science and Information Technology Florida State University Tallahassee FL 32306 Abstract Linearization of 1-D Euler equations about a discontinuous solution is discussed from both the theoretical and numerical point of view. Estimates for the norm of the solution of the linearized system are shown to be valid for the case presented. Numerically, the linearization is performed following the guidelines of tangent linear model and sensitivities with respect to a flow parameter are computed, being in better agreement with the analytical value when compared with previously reported results. Key words: sensitivity, Euler equations, discontinuity, linearization, sensitivity estimates, tangent linear model 1 Introduction Sensitivities (which are derivatives of the variables or cost functionals that describe the model with respect to parameters that determine the behavior of the model –e.g. initial conditions, boundary conditions, shape parameters etc.) were extensively studied in the last decades for problems involving continuous functions. Research was performed also in the presence of discontinuities but many questions in this area remain yet unanswered. ? Corresponding author. Email addresses: chome@math.fsu.edu (Chris Homescu), navon@csit.fsu.edu ( I.M. Navon). Preprint submitted to Elsevier Science 7 September 2001 Sensitivity analysis in the case of a model with discontinuities was applied in areas like fluid dynamics, aerodynamics, chemistry, financial analysis, meteo- rology or environmental studies, to name but a few. Studies include shape optimization for fluids (Burgreen and Baysal [1], New- man et al. [2], Taylor et al. [3], Mohammadi and Pironneau [4]), noise analysis and optimization of electronic circuits (Nguyen et al. [5]), control of contami- nant releases in rivers (Piasecki and Kato