  Multisymplectic variational integrators are structure preserving numerical
schemes especially designed for PDEs derived from covariant spacetime Hamilton
principles. The goal of this paper is to study the properties of the temporal
and spatial discrete evolution maps obtained from a multisymplectic numerical
scheme. Our study focuses on a 1+1 dimensional spacetime discretized by
triangles, but our approach carries over naturally to more general cases. In
the case of Lie group symmetries, we explore the links between the discrete
Noether theorems associated to the multisymplectic spacetime discretization and
to the temporal and spatial discrete evolution maps, and emphasize the role of
boundary conditions. We also consider in detail the case of multisymplectic
integrators on Lie groups. Our results are illustrated with the numerical
example of a geometrically exact beam model.
