NUMERICAL METHODS FOR ONE-DIMENSIONAL REACTION-DIFFUSION EQUATIONS ARISING IN COMBUSTION THEORY

DOI: 10.1615/AnnualRevHeatTransfer.v1.60
pages 150-261

Juan I. Ramos
Escuela de Ingenierias Industriales, Universidad de Malaga, Dr. Ortiz Ramos, s/n 29071 Malaga, Spain

Abstract

A review of numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory is presented. The methods reviewed include explicit, implicit, quasilinearization, time linearization, operator-splitting, random walk and finite-element techniques and methods of lines. Adaptive and non-adaptive procedures are also reviewed. These techniques are applied first to solve two model problems which have exact traveling wave solutions with which the numerical results can be compared. This comparison is performed in terms of both the wave profile and computed wave speed. It is shown that the computed wave speed is not a good indicator of the accuracy of a particular method. A fourth-order time-linearized, Hermitian compact operator technique is found to be the most accurate method for a variety of time and space sizes. The accuracy of this time-linearization technique degrades as large time steps are used in the calculations. Adaptive and moving finite-difference and finite-element methods are shown to be very accurate techniques which do not require as much computer time as non-adaptive methods.
The solution of steady state flame propagation problems can be achieved more quickly with boundary value techniques and Newton method if a good initial estimate is available. Otherwise, the time-dependent equations may need to be solved until an estimate which lies in the domain of convergence of Newton's method is obtained. Numerical methods have been used in recent years to develop and validate detailed reaction mechanisms for the combustion of fuels. They have also been used to analyze the sensitivity of a particular reaction mechanism to variations in the initial and boundary conditions, transport properties and reaction data. Changes in nondimensional parameters can yield multiple steady states, oscillations and traveling wave solutions. A brief review of the methods employed to perform sensitivity and bifurcation analyses is also presented. Some of the techniques presented in the paper have also been applied to ignition studies and two-phase reacting flow problems. These studies are briefly reviewed in this paper.