  We present a structure preserving discretization of the fundamental spacetime
geometric structures of fluid mechanics in the Lagrangian description in 2D and
3D. Based on this, multisymplectic variational integrators are developed for
barotropic and incompressible fluid models, which satisfy a discrete version of
Noether theorem. We show how the geometric integrator can handle regular fluid
motion in vacuum with free boundaries and constraints such as the impact
against an obstacle of a fluid flowing on a surface. Our approach is applicable
to a wide range of models including the Boussinesq and shallow water models, by
appropriate choice of the Lagrangian.
