Asymptotics and simulation of singular phenomena

点击翻译按钮获取中文摘要

The qualitative study of nonlinear dynamical systems is crucial in understanding many phenomena in science and technology. The study of such behaviour has received much attention in model problems arising from numerous diverse areas including combustion theory, nonlinear optics, plasma physics, hydrodamics, mathematical biology and differential geometry. In these systems the effect of nonlinear reaction terms may overcome spatial diffusion leading to singularity formation wherein the solution becomes unbounded in some norm at a finite time.  These singularities may indicate model breakdown where there is important dynamic behaviour on scales not captured (eg. extreme focussing) or they may correspond to a qualitative change in the physical state (eg. explosion).  A careful resolution of the spatio-temporal dynamics near such singular behaviour is crucial to understanding many pure and applied PDE problems.  My research involves developing asymptotic and numerical approaches to studying this delicate phenomenon. Specifically, the project includes: 1) Analysis of finite-time singularities in nonlinear PDEs arising in physics and differential geometry; 2) Analysis and implementation of moving mesh methods in multiple dimensions; 3) Analysis and implementation of scale-invariant moving mesh methods for blow-up problems.