  We present a compatible space-time hybridizable/embedded discontinuous
Galerkin discretization for nonlinear free-surface waves. We pose this problem
in a two-fluid (liquid and gas) domain and use a time-dependent level-set
function to identify the sharp interface between the two fluids. The
incompressible two-fluidd equations are discretized by an exactly mass
conserving space-time hybridizable discontinuous Galerkin method while the
level-set equation is discretized by a space-time embedded discontinuous
Galerkin method. Different from alternative discontinuous Galerkin methods is
that the embedded discontinuous Galerkin method results in a continuous
approximation of the interface. This, in combination with the space-time
framework, results in an interface-tracking method without resorting to
smoothing techniques or additional mesh stabilization terms.
