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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. I. Introduction While much work has been done in the field of steady adjoint approach for optimal control and inverse design problems using traditional CFD methods 1–4 such as finite volume methods, the current trend and demand for more time-accurate and spatial-accurate methods entail further development of the adjoint-based approach. Recent research has seen the combination of the unsteady adjoint method with the high order methods such as the Discontinous Galerkin (DG) method. 5–7 Method of this type can be a significant aid to the field of aeroacoustic design and optimization, since the capture of acoustic signiture often requires the time-accuracy and spatial-accuracy of the high order methods which have very small numerical dissipations. In this study, we investigate the formulation of unsteady adjoint approach based on high-order spectral difference (SD) method, and explore methods that can lead to better computational and optimization efficiency. We consider the problems of the optimal control of the advection and Burger’s equations, by treating the initial conditions as the control inputs, and matching the final solutions with design target profiles.

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