Unsteady Adjoint Method for the Optimal Control of Advection and Burger's Equations Using High-Order Spectral Difference Method

In this study, we investigate the method for adjoint-based optimal control of linear and non-linear equations. In particular, we are interested in the formulation of the unsteady adjoint method based on the high-order spectral difference method. For the linear equation, we consider the simple 1D advection equation. For the non-linear equation, we consider the 1D viscous Burger’s equation. In both cases, the inverse design of the target solution is achieved through control of the initial condition. The focus of the study is the formulation of the unsteady adjoint method using the high-order spectral difference method. The paper demonstrates that the minimal numerical dissipation inherent in the high order method is very beneficial for adjoint type problem where backward integration in time is involved. The combination of adjoint approach and high order method will lead to an useful tool for optimization in the field of aeroacoustic simulation and design.