Numerical Solutions for Transient and Nearly Periodic Waves in Shallow Water

This paper presents a study of several numerical methods for solving transient (time-dependent) model equations for waves in shallow water by means of the method of lines. The physical models studied include a time dependent mild-slope equation for narrow-banded linear waves in intermediate water depth, and nonlinear models for weakly dispersive long waves in Boussinesq and Green-Naghdi form. The models treat spatial dependence using second and fourth-order accurate finite-differences, and time integration is accomplished using a variety of methods including Euler predictor-corrector, fourth-order Runge-Kutta, and the Bulirsch-Stoer method using a modified midpoint scheme with polynomial extrapolation and adaptive step size.