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.
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