| Traditional numerical techniques to solve time-dependent partial differential equations (PDEs) integrate on a uniform spatial grid that is kept fixed on the entire time interval. If the solutions have regions of high spatial activity, a standard fixed-grid technique is computationally inefficient, since to afford an accurate numerical approximation, it should contain, in general, a very large number of grid points. The grid on which the PDE is discretized then needs to be locally refined. Moreover, if the regions of high spatial activity are moving in time, like for steep moving fronts in reaction-diffusion or hyperbolic equations, then techniques are needed that also adapt (move) the grid in time. In the realm of adaptive techniques for time-dependent PDEs we can, roughly spoken, distinguish between two classes of methods: $h$- and $r$-refinement methods. During this series of lectures we are going to learn about the latter class of adaptive methods. Moving mesh methods, also denoted by the term $r$-refinement ($r$e-distribute or $r$e-locate), have the special feature to move the spatial grid continuously and automatically in the space-time domain while the discretization of the PDE and the moving-grid procedure are intrinsically coupled. Moving-grid techniques use a fixed number of grid points, without need of interpolation and let the grid points dynamically move with the underlying feature of the PDE (wave, pulse, front, ...). |