  We present a systematic derivation of the algorithms required for computing
the gradient and the action of the Hessian of an arbitrary misfit function for
large-scale parameter estimation problems involving linear time-dependent PDEs
with stationary coefficients. These algorithms are derived using the adjoint
method for time-stepping schemes of arbitrary order and are therefore
well-suited for distributed parameter estimation problems where the forward
solution needs to be solved to high accuracy. Two examples demonstrate how
specific PDEs can be prepared for use with these algorithms. A numerical
example illustrates that the order of accuracy of higher-order time-stepping
schemes is inherited by their corresponding adjoint time-stepping schemes and
misfit gradient computations.
