  We present a novel solution procedure for initial boundary value problems.
The procedure is based on an action principle, in which coordinate maps are
included as dynamical degrees of freedom. This reparametrization invariant
action is formulated in an abstract parameter space and an energy density scale
associated with the space-time coordinates separates the dynamics of the
coordinate maps and of the propagating fields. Treating coordinates as
dependent, i.e. dynamical quantities, offers the opportunity to discretize the
action while retaining all space-time symmetries and also provides the basis
for automatic adaptive mesh refinement (AMR). The presence of unbroken
space-time symmetries after discretization also ensures that the associated
continuum Noether charges remain exactly conserved. The presence of coordinate
maps in addition provides new freedom in the choice of boundary conditions. An
explicit numerical example for wave propagation in $1+1$ dimensions is
provided, using recently developed regularized summation-by-parts finite
difference operators.
