A High‐Order Hybrid‐Spectral Incompressible Navier–Stokes Model for Non‐Linear Water Waves

ABSTRACT 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 non‐linear 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 fourth‐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 results in the emergence of a Poisson pressure problem that constitutes 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 ‐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 ‐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.