  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.
