Numerical simulation of singular solutions of the Generalized Korteweg - de Vries Equation

A fully discrete approximation to the solution of the Generalized Korteweg - de Vries Equation is presented and analyzed. The presented numerical method is aimed at approximating solutions developing blow-up singularities. The spatial discretization is based on polynomial spline, adaptive finite elements and the temporal discretization is achieved by a variable step Crank-Nicolson scheme. The numerical scheme employs a spatial refinement strategy which allows the mesh to be refined locally and without compromising the overall quality of the mesh. Several numerical experiments concerning the formation of singularities are presented.