This monograph deals with the structure, generation and stability of flames from a mathematical point of view. It uses a specific mathematical approach to provide a unified theoretical description of fundamental flame phenomena. Its importance stems from the fact that it provides the first clear evidence that combustion can be legitimately treated as a mathematical science as well as an empirical one. The book will be of interest to researchers in combustion, fluid mechanics and applied mathematics, as well as to graduate students taking advanced courses in these areas.

Combustion Theory delves deeper into the science of combustion than most other texts and gives insight into combustions from a molecular and a continuum point of view. The book presents derivations of the basic equations of combustion theory and contains appendices on the background of subjects of thermodynamics, chemical kinetics, fluid dynamics, and transport processes. Diffusion flames, reactions in flows with negligible transport and the theory of pre-mixed flames are treated, as are detonation phenomena, the combustion of solid propellents, and ignition, extinction, and flamibility pehnomena.

In this paper we describe the numerical analysis underlying our efforts to develop an accurate and reliable code for simulating flame propagation using complex physical and chemical models. We discuss our spatial and temporal discretization schemes, which in our current implementations range in order from two to six. In space we use staggered meshes to define discrete divergence and gradient operators, allowing us to approximate complex diffusion operators while maintaining ellipticity. Our temporal discretization is based on the use of preconditioning to produce a highly efficient linearly implicit method with good stability properties. High order for time accurate simulations is obtained through the use of extrapolation or deferred correction procedures. We also discuss our techniques for computing stationary flames. The primary issue here is the automatic generation of initial approximations for the application of Newton's method. We use a novel time-stepping procedure, which allows the dynamic updating of the flame speed and forces the flame front towards a specified location. Numerical experiments are presented, primarily for the stationary flame problem. These illustrate the reliability of our techniques, and the dependence of the results on various code parameters. Hagstrom, Thomas and Radhakrishnan, Krishnan and Zhou, Ruhai Glenn Research Center NASA/CR-1999-209305, NAS 1.26:209305, AIAA Paper 98-3246, ICOMP-99-07, E-11826