Mode-based derivation of adjoint equations - a lazy mans approach

A method to calculate the adjoint solution for a large class of partial differential equations is discussed. It differs from the known continuous and discrete adjoint, including automatic differentiation. Thus, it represents an alternative, third method. It is based on a modal representation of the linearized operator of the governing (primal) system. To approximate the operator an extended version of the Arnoldi factorization, the dynamical Arnoldi method (DAM) is introduced. The DAM allows to derive approximations for operators of non-symmetric coupled equations, which are inaccessible by the classical Arnoldi factorization. The approach is applied to the Burgers equation and to the Euler equations on periodic and non-periodic domains. Finally, it is tested on an optimization problem.