  We present a systematic methodology to develop high order accurate numerical
approaches for linear advection problems. These methods are based on evolving
parts of the jet of the solution in time, and are thus called jet schemes.
Through the tracking of characteristics and the use of suitable Hermite
interpolations, high order is achieved in an optimally local fashion, i.e. the
update for the data at any grid point uses information from a single grid cell
only. We show that jet schemes can be interpreted as advect-and-project
processes in function spaces, where the projection step minimizes a stability
functional. Furthermore, this function space framework makes it possible to
systematically inherit update rules for the higher derivatives from the ODE
solver for the characteristics. Jet schemes of orders up to five are applied in
numerical benchmark tests, and systematically compared with classical WENO
finite difference schemes. It is observed that jet schemes tend to possess a
higher accuracy than WENO schemes of the same order.
