Method of lines study of nonlinear dispersive waves

In this study, we consider partial differential equation problems describing nonlinear wave phenomena, e.g., a fully nonlinear third order Korteweg-de Vries (KdV) equation, the fourth order Boussinesq equation, the fifth order Kaup–Kupershmidt equation and an extended KdV5 equation. First, we develop a method of lines solution strategy, using an adaptive mesh refinement algorithm based on the equidistribution principle and spatial regularization techniques. On the resulting highly nonuniform spatial grids, the computation of high-order derivative terms appears particularly delicate and we focus attention on the selection of appropriate approximation techniques. Finally, we solve several illustrative problems and compare our computational approach to conventional solution techniques.