Overview

Nowadays, adaptive methods are widely used to solve partial differential equations which involve large solution variations like shock waves, boundary layers, etc. These techniques can now be considered as fully mature. Indeed, they have successfully demonstrated their ability in significantly reducing the computational cost of computer simulations while simultaneously preserving or even improving the accuracy of the numerical solutions. The main idea is to adapt the mesh points locally in order to concentrate them in the regions where the gradient of the solution rapidly varies (see Chapters 21 and 22). However, if the regions of interest are moving or evolving in time, such as fronts or shocks in hyperbolic equations, these techniques need to be modified to adapt the mesh with respect to time [Alauzet et al. 2007].

Regarding the numerical simulation of time-dependent PDE problems, mesh adaptation methods can be split into two categories, static or dynamic. The first class, corresponding basically to h-refinement techniques, consists of adapting the mesh by possibly adding nodes and updating node positions and interpolating variables from the old mesh to the new mesh, all these operations being performed at discrete time levels. Obviously, the principal attraction of these methods lies in their simplicity and, to some extent, their robustness in dealing with an arbitrary number of phenomena concurrently. However, a well known artifact is often noticed when dealing for instance with hyperbolic PDEs, the numerical dispersion. In addition, in order to preserve the time stepping accuracy, small steps...