Linear finite-element numerical techniques for combustion problems requiring variable step size

Combustion problems frequently pose situations in which the dependent variables (temperature, species concentrations, etc.) vary rapidly in some local domain and much more slowly throughout the remaining region of interest. Such situations arise for both fluid-mechanical (e.g., boundary-layer behavior) and chemical reasons (i.e., Arrhenius- type temperature dependence of chemical reaction rates). In any case, they present a particular difficulty in the numerical solution of the governing conservation equations: if a fixed space-step is to be used, its magnitude is dictated by stability and/or accuracy requirements of the limited but rapidly varying domain. Outside this narrow domain, many more space steps are taken than are required; computing costs are thus magnified. Various means to avoid this difficulty have been used in the past, but none has proved generally satisfactory or convenient. Finite-element methods are substantially more convenient than the classical finite-difference techniques for approximating spatial dependencies in certain combustion problems. We discuss only linear finite-element (LFE) methods here. A discussion of this and more general finite-element methods can be found in several sources [1-3]; we briefly outline the application of the LFE method and illustrate its advantages