  We present an iterative coupling scheme for the numerical approximation of
the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove
its convergence in the Banach space setting for an abstract semi-discretization
in time that allows the application of the family of diagonally implicit
Runge-Kutta methods. Recasting the semi-discrete solution as the minimizer of a
properly defined energy functional, the proof of convergence uses its
alternating minimization. The scheme is closely related to the undrained split
for the quasi-static Biot system.
