  We propose a time-adaptive predictor/multi-corrector method to solve
hyperbolic partial differential equations, based on the generalized-$\alpha$
scheme that provides user-control on the numerical dissipation and second-order
accuracy in time. Our time adaptivity uses an error estimation that exploits
the recursive structure of the variable updates. The predictor/multicorrector
method explicitly updates the equation system but computes the residual of the
system implicitly. We analyze the method's stability and describe how to
determine the parameters that ensure high-frequency dissipation and accurate
low-frequency approximation. Subsequently, we solve a linear wave equation,
followed by non-linear finite strain deformation problems with different
boundary conditions. Thus, our method is a straightforward, stable and
computationally efficient approach to simulate real-world engineering problems.
Finally, to show the performance of our method, we provide several numerical
examples in two and three dimensions. These challenging tests demonstrate that
our predictor/multicorrector scheme dynamically adapts to sudden energy
releases in the system, capturing impacts and boundary shocks. The method
efficiently and stably solves dynamic equations with consistent and
under-integrated mass matrices conserving the linear and angular momenta as
well as the system's energy for long-integration times.
