Adaptive collocation methods for the numerical solution of differential models

A PDE integration algorithm that associates a Method of Lines (MOL) strategy based on finite differences or high resolution space discretizations, with a collocation strategy based on increasing level one or two-dimensional dyadic grids is presented. It reveals potential either as a grid generation procedure for predefined steep localised functions, and as an integration scheme for moving steep gradient PDE problems, namely 1D and 2D Burgers equations. Therefore, it copes satisfactorily with an example characterized by a steep 2D travelling wave and an example characterised by a forming steep travelling shock, which confirms its flexibility in dealing with diverse types of problems, with reasonable demands of computational effort