  Variational integrators for Lagrangian dynamical systems provide a systematic
way to derive geometric numerical methods. These methods preserve a discrete
multisymplectic form as well as momenta associated to symmetries of the
Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation
of variational integrators is the existence of a variational formulation for
the considered problem. Even though for a large class of systems this
requirement is fulfilled, there are many interesting examples which do not
belong to this class, e.g., equations of advection-diffusion type frequently
encountered in fluid dynamics or plasma physics. On the other hand, it is
always possible to embed an arbitrary dynamical system into a larger Lagrangian
system using the method of formal (or adjoint) Lagrangians. We investigate the
application of the variational integrator method to formal Lagrangians, and
thereby extend the application domain of variational integrators to include
potentially all dynamical systems. The theory is supported by physically
relevant examples, such as the advection equation and the vorticity equation,
and numerically verified. Remarkably, the integrator for the vorticity equation
combines Arakawa's discretisation of the Poisson brackets with a symplectic
time stepping scheme in a fully covariant way such that the discrete energy is
exactly preserved. In the presentation of the results, we try to make the
geometric framework of variational integrators accessible to non specialists.
