Optimization of non-hydrostatic Euler model for water waves

The distribution of variables in the water column controls the dispersion properties of non-hydrostatic models. Because solving the Poisson equation is the most time consuming part of a non-hydrostatic model, it is highly desirable to reduce the number of unknowns in the water column by placing them at optimal locations. The paper presents the analytical dispersion relationship of a non-hydrostatic Euler model for water waves. The phase speed of linear waves simulated by the semi-discretized Euler model can be expressed as a rational polynomial function of the dimensionless water depth, kh, and the thicknesses of layers encompassing the water column become optimizable parameters in this function. The dispersion error is obtained by comparing this phase speed against the exact solution based on the linear wave theory. It is shown that for a given dispersion error (e.g. 1%), the range of kh can be extended if the layer thicknesses are optimally selected. The optimal two- and three-layer distributions for the aforementioned Euler model are provided. The Euler model with the optimized layer distribution shows good linear dispersion properties up to kh ≈ 9 with two layers, and kh ≈ 49.5 with three layers. The derived phase speed was tested against both numerical and exact solutions of standing waves for various cases. Excellent agreement was achieved. The model was also tested using the fifth-order Stokes theory for nonlinear standing waves. The phase speed of nonlinear waves follows a similar trend of the derived phase speed although it deviates proportionally with the increase of wave steepness ka0. Thus, the optimal layer distribution can also be applied to nonlinear waves within a range of ka0 and kh. The optimization method is applicable to other non-hydrostatic Euler models for water waves.