Optimal sponge layer for water waves numerical models

We present the system of linear equations to be solved to evaluate the reflection coefficient at a sponge layer boundary with damping forces ρf_SL=−ρβu, where β is the sponge layer function. The case of 2D waves is discussed in details and different sponge layer functions are proposed. The linear system is solved using the finite element method (FEM) at low computational cost compared to the target simulations. This method should enable the design of efficient sponge layer for any full Navier–Stokes solvers. The linear model results solved with a FEM solver are compared here to SPH simulations with good agreement. The linear model is then used to determine the reflection coefficient for different power sponge functions, length and dissipation coefficient using the FEM solver. The linear model solved by a simple FEM solver can be used to evaluate the suitable dissipation coefficients for waves ranging from shallow to deep water for a given sponge layer length. The coefficient depends on the non-dimensional frequency for Ω=ωd/g<2. A table of suitable parameters for different sponge layer functions is provided over a large range of non-dimensional frequency and sponge layer length. A 3D application of waves is also presented with 45° incidence angle.