Time Accuracy of Pressure Methods for Nonhydrostatic Free-Surface Flows

We extend two fractional-step pressure methods (pressure projection and pressure correction) to nonhydrostatic free-surface flows by employing them to discretize the nonhydrostatic pressure on a staggered grid and applying a semi-implicit time discretization to the free-surface elevation. The time accuracy and efficiency for each pressure method is investigated in detail. Using a sloshing free-surface wave as a test case, we show analytically and numerically that the correction method is always second-order accurate in time, while the projection method is second-order accurate in time only if the nonhydrostatic pressure is included in the depth-averaged continuity equation through the use of an iterative procedure. We demonstrate that this convergence behavior holds for the linearized equations.