toBlow - up Problems for PDEs

In this paper, we give a short review of the moving mesh methods based upon moving mesh PDEs, including a brief description of a new moving collocation method. Then we carry out a numerical study of a class of PDEs describing blow-up of solutions of combustion problems. The study demonstrates the usefulness of these methods as a tool for the analyst. Speciically, we use the numerical method as an experimental approach to conjecture the solution proole for blow-up problems and verify the conclusions obtained from a formal (and not necessarily rigorous) analysis of these PDEs. It is now widely recognized that when solving partial diierential equations (PDEs) which have components of the solution which vary greatly over small length scales some form of adaptivity is necessary. In this paper we review some adaptive methods, based upon the method of lines for solving parabolic PDEs of the form u t = F(u; u x ; u xx); 0 < x < 1 (1:1) In particular, we consider their behavior for problems that form singularities in a nite time.