We will discuss an adaptive grid method based upon a moving mesh approach for solving time dependent PDEs. The approach is based upon a moving mesh PDE (MMPDE) formulation. In higher dimensions, the MMPDE is derived from a heat flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps. Geometrical interpretations are given for the heat equation and functional, and basic properties of this MMPDE are discussed.
The method is relatively simple and easy to program. Numerical examples are presented where it is used both for mesh generation and for solving time dependent parabolic PDEs. The results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh smoothness (usually measured by the mesh orthogonality).