An Adjoint Method using Fully Implicit Runge-Kutta Schemes for Optimization of Flow Problems

The fully discrete adjoint equations and corresponding adjoint method are derived for unsteady PDE-constrained optimization problems. Specifically, we consider conservation laws on deforming domains that are temporally discretized by high-order fully implicit Runge-Kutta (IRK) schemes. Through a change of variables, the linear systems arising in the primal and dual problem are transformed, leading to computationally cheaper systems that compare competitively with those derived from diagonally implicit RungeKutta (DIRK) schemes. Quantities of interest that take the form of space-time integrals are discretized in a solver-consistent manner. Our fully discrete, IRK adjoint method is used to compute exact gradients of quantities of interest with respect to the optimization parameters. These quantities of interest and their gradients are used for gradient-based PDE-constrained optimization. Our implementation of this IRK adjoint method is tested by computing the energetically optimal trajectory of a 2D airfoil in flow governed by the compressible Navier-Stokes equations. We also analyze the parallel performance of our IRK adjoint method and the DIRK adjoint method, showing that our implementation is computationally comparable.

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