Qin Sheng 

(Department of Mathematics, Dayton University) 

Recent Advances in Adaptive and Adaptive-Splitting Methods for
Nonlinear Quenching-Combustion Problems

Abstract

The original mathematical study of quenching-combustion equations can be traced back to Kawarada in 1975. In his study of an one dimensional heat equation with a singular source term, Kawarada observed that there exists a particular value a*>0 such that if a, the length of the spatial interval, is less than a*, then the solution of the equation exists globally. However, if a >= a* then there exists a finite time T(a), such that the solution of the differential equation stops existing at T(a). Kawarada referred this as quenching. Most important quantities used in quenching problem characterizations include critical value a* and quenching time T(a).

Quenching phenomena are common and distinguished for their important physical, engineering interpretations and unique applications in the combustion engin industry. They may serve as an indicator of the steady and unsteady combustion processes, or burning and explosion of the rocket fuel. The solution of quenching type partial differential equations also play a significant role in the theory of ecology and environmental studies, and in particular, in the prediction and control of pipeline decays.

There have been considerable developments in quenching studies in recent years, in particular due to contributions of Acker, Chan, Deng, Levine, Olmstead and Walter. Different types of adaptive schemes dealing with quenching singularities are also developed for better computations. In this talk, we will focus on a number of interesting linearly implicit adaptive difference methods for solving quenching type differential equations. Special attention will be paid to monotone properties of the numerical solution. Further discussions on adaptive methods for quenching problems, such as the solution deviation and correction, will be given.
 
Last updated by fass@amadeus.math.iit.edu  on 08/29/03