Numerical Methods for Solving Reaction-Diffusion Problems

摘要

A reaction-diffusion model problem containing many of the characteristics of more general combustion problems was formulated in a dimensionless form and used to assess the accuracy and efficiency of six numerical solution procedures. Exothermic heat release and chemical conversion of the various species are described by Arrhenius-type reactions. Both one- and two-step chemical kinetic processes were considered. The numerical solution procedures evaluated included two methods of lines techniques, three schemes which use pointwise integration of reaction terms in conjunction with various diffusion models and a locally linearized, block tridiagonal, implicit method. The efficiency of both second and fourth order spatial approximations with each of the six solution procedures was evaluated and a fourth order correct approximation for each was chosen for use in subsequent comparison tests. All six methods were found to be adequate in terms of predicting flame speed, temperature profiles and the spatial distribution of the various species including trace species in amounts on the order of parts per million. However, the linearized block tridiagonal procedure was found to be most efficient over a wide range of reaction rates including cases with considerable stiffness. In many cases this method required only one sixth as much computer time as the most competitive of the other five procedures. In none of the cases examined was another procedure found to be more efficient. Extension to combustion problems in a gaseous medium and two-dimensional geometries indicated that most of the advantages exhibited in solving the model problem will be retained when this procedure is applied to problems of a more complex nature. 20 figures, 50 tables. (ERA citation 04:014206)