  We present a new high-order accurate computational fluid dynamics model based
on the incompressible Navier-Stokes equations with a free surface for the
accurate simulation of nonlinear and dispersive water waves in the time domain.
The spatial discretization is based on Chebyshev polynomials in the vertical
direction and a Fourier basis in the horizontal direction, allowing for the use
of the fast Chebyshev and Fourier transforms for the efficient computation of
spatial derivatives. The temporal discretization is done through a generalized
low-storage explicit 4th order Runge-Kutta, and for the scheme to conserve mass
and achieve high-order accuracy, a velocity-pressure coupling needs to be
satisfied at all Runge-Kutta stages. This result in the emergence of a Poisson
pressure problem that constitute a geometric conservation law for mass
conservation. The occurring Poisson problem is proposed to be solved
efficiently via an accelerated iterative solver based on a geometric
$p$-multigrid scheme, which takes advantage of the high-order polynomial basis
in the spatial discretization and hence distinguishes itself from conventional
low-order numerical schemes. We present numerical experiments for validation of
the scheme in the context of numerical wave tanks demonstrating that the
$p$-multigrid accelerated numerical scheme can effectively solve the Poisson
problem that constitute the computational bottleneck, that the model can
achieve the desired spectral convergence, and is capable of simulating
wave-propagation over non-flat bottoms with excellent agreement in comparison
to experimental results.
