A Continuous Sensitivity Equation of Arbitrary High Order
Corinne Belley, Alexander Hay, Dominique H. Pelletier · 22nd AIAA Computational Fluid Dynamics Conference · 2015
We present an approach to automatically generate and solve the flow sensitivities with respect to a given single parameter up to an arbitrary order n. We use the Newton multinomial theorem to automatically derive the set of terms constituting the sensitivity equations of any order. Hence, given the flow equations at hand (Navier-Stokes, RANS, Burgers, etc), our methodology automatically produces the corresponding equations for the flow sensitivities of an arbitrary high order n. In our approach, the flow and sensitivity equations are not calculated by different solvers resulting from different source codes. Rather, we extend an existing flow solver by adding an extra loop over the sensitivity order (i.e. from 0 to n, the 0 order flow sensitivity being the flow itself) on top of the main solution procedure. Thus, during the execution of the loop the first iteration computes the flow as before and the next iterations compute the flow sensitivities up to the requested order n. We present the necessary generic data structure to do so. The verification of the flow-and-sensitivity solver is performed by the method of the manufactured solution. The computed sensitivities are validated by comparison to sensitivities obtained by second-order finite-differences. Finally, we examine the ability of high-order Taylor series expansions in parameter space to approximate flow solutions over a wide range of parameter values.