On the forward in time solution of the unsteady adjoint equations

A new algorithm for the solution of the unsteady adjoint equations is proposed in this paper aiming at overcoming the excessive computational cost and memory requirements of the conventional adjoint approach for the optimization of unsteady problems in computational mechanics. The proposed algorithm consists of two phases: the first one is based on the backward in time solution of the adjoint equations based on average in time state quantities and the second one solves the exact unsteady adjoint equations forward in time, based on the initial adjoint condition derived from the averaging technique of the first phase. The proposed algorithm is compared to other approaches in the case of the 1D unsteady Burgers equation with non-smooth source terms which resembles the modelling of unsteady turbulent flows with very fine temporal discretization modeled using LES, DES or DNS methods. The adjoint variables are computed with acceptable accuracy at a cost of four times solving the unsteady state equations with negligible additional memory requirements