A two-dimensional σ-transform based finite element method for nonlinear water waves

A finite element model utilizing the sigma-transform is presented to address the two-dimensional fully nonlinear potential flow problem. The Euler-Zakharov equation, discretized with Hermite elements, serves as the timedependent free surface boundary condition. Higher-order accuracy is achieved through polynomial approximations in the vertical dimension. Temporal discretization employs the explicit Runge-Kutta method with adaptive step size. Convergence studies and stability assessments for long-time simulations are conducted. Model accuracy is validated through two cases of wave propagation over a trapezoidal bar. The results demonstrate high efficiency and rapid convergence under finite water depth conditions. Comparison with experimental data confirms the method's accuracy in predicting the propagation and dispersion of nonlinear water waves.