  We consider the dynamics of a parabolic and a hyperbolic equation coupled on
a common interface and develop time-stepping schemes that can use different
time-step sizes for each of the subproblems. The problem is formulated in a
strongly coupled (monolithic) space-time framework. Coupling two different step
sizes monolithically gives rise to large algebraic systems of equations where
multiple states of the subproblems must be solved at once. For efficiently
solving these algebraic systems, we inherit ideas from the partitioned regime
and present two decoupling methods, namely a partitioned relaxation scheme and
a shooting method. Furthermore, we develop an a posteriori error estimator
serving as a mean for an adaptive time-stepping procedure. The goal is to
optimally balance the time step sizes of the two subproblems. The error
estimator is based on the dual weighted residual method and relies on the
space-time Galerkin formulation of the coupled problem. As an example, we take
a linear set-up with the heat equation coupled to the wave equation. We
formulate the problem in a monolithic manner using the space-time framework. In
numerical test cases, we demonstrate the efficiency of the solution process and
we also validate the accuracy of the a posteriori error estimator and its use
for controlling the time step sizes.
