A Fourier-Chebyshev collocation method for the shallow water equations including shoreline runup

A solution method for the shallow water equations governing wave motions in the nearshore environment is presented. Spatial derivatives contained in these equations are computed using spectral collocation methods. A high-order time integration scheme is used to compute the time evolution of the velocities and water surface elevation given initial conditions. The model domain extends from the shoreline to a desired distance offshore and is periodic in the longshore direction. Properly posed boundary conditions for the governing equations are discussed. A curvilinear moving boundary condition is incorporated at the shoreline to account for wave runup. An absorbing-generating boundary is constructed offshore. The boundary treatments are tested using analytical and numerical results. The developed method is applied to the prediction of neutral stability boundaries and equilibrium amplitudes of subharmonic edge waves. Numerical results are compared to weakly nonlinear theory and are found to reproduce the theory very well.