Renormalization and the renormalization group (RG) were originally developed by physicists attempting to understand the divergent terms in perturbation theory and the short distance behaviour of quantum electrodynamics. During the last few years, these methods have been used to study the divergent terms in perturbation theory and the long time behaviour of a variety of partial differential equations. Problems studied include similarity solutions, especially intermediate asymptotics of the second kind (Barenblatt classification), and travelling waves. Most recently, singular perturbation problems have been treated, with particular attention paid to multiple-scale analysis, boundary layers and WKB, and matched asymptotics.
The RG works from the inner expansion alone, and never requires asymptotic matching. The RG method starts from a regular perturbation expansion in the small parameter, and automatically generates an asymptotic sequence without requiring the user to make insightful guesses as to the presence of "unexpected" powers, logarithms, etc. The RG-generated uniform approximation is practically more useful than that generated by matched asymptotics, even when extended to values of the small parameter of order unity. It seems, then, that the RG is widely applicable and can be used mechanically.
All of these results are formal, and require significant mathematical investigation in order to be justified and properly understood.
This is work performed in collaboration with Yoshitsugu Oono, L.-Y. Chen, O. Martin, F. Liu.