Adaptive methods of lines for one‐dimensional reaction‐diffusion equations
Authors: Ramos J. I. 
Published: August 2005

Adaptive and non‐adaptive finite difference methods are used to study one‐dimensional reaction‐diffusion equations whose solutions are characterized by the presence of steep, fast‐moving flame fronts. Three non‐adaptive techniques based on the methods of lines are described. The first technique uses a finite volume method and yields a system of non‐linear, first‐order, ordinary differential equations in time. The second technique uses time linearization, discretizes the time derivatives and yields a linear, second‐order, ordinary differential equation in space, which is solved by means of three‐point, fourth‐order accurate, compact differences. The third technique takes advantage of the disparity in the time scales of the reaction and diffusion processes, splits the reaction‐‐diffusion operator into a sequence of reaction and diffusion operators and solves the diffusion operator by means of either a finite volume method or a three‐point, fourth‐order accurate compact difference expression. The non‐adaptive methods of lines presented in this paper may use equaliy or non‐equally spaced fixed grids and require a large number of grid points to solve accurately one‐dimensional problems characterized by the presence of steep, fast‐moving fronts. Three adaptive methods for the solution of reaction‐diffusion equations are considered. The first adaptive technique is static and uses a subequidistribution principle to determine the grid points, avoid mesh tangling and node overtaking and obtain smooth grids. The second adaptive technique is dynamic, uses an equidistribution principle with spatial and temporal smoothing and yields a system of first‐order, non‐linear, ordinary differential equations for the grid point motion. The third adaptive technique is hybrid, combines some features of static and dynamic methods, and uses a predictor‐corrector strategy to predict the grid and solve for the dependent variables, respectively. The three adaptive techniques presented in this paper use physical co‐ordinates and may employ finite volume or three‐point, compact methods. The adaptive and non‐adaptive finite difference methods presented in the paper are used to study a decomposition chemical reaction characterized by a scalar, one‐dimensional reaction‐diffusion equation, the propagation of a one‐dimensional, confined, laminar flame in Cartesian co‐ordinates and the Dwyer‐Sanders model of one‐dimensional flame propagation. It is shown that the adaptive moving method presented in this paper requires a smaller number of grid points than adaptive static, adaptive hybrid and non‐adaptive methods. The adaptive hybrid method requires a smaller time step than adaptive static techniques, due to the lag between the grid prediction and the solution of the dependent variables. Non‐adaptive methods of lines may yield temperature oscillations in front of and behind the flame front if Crank‐Nicolson techniques are used to evaluate the time derivatives. Fourth‐order accurate methods of lines in space yield larger temperature oscillations than second‐order accurate methods of lines, and the magnitude of these oscillations decreases as the time step is decreased. It is also shown that three‐point, fourth‐order accurate discretizations of the spatial derivatives require the same number of grid points as second‐order accurate, finite volume methods, in order to resolve accurately the structure of steep, fast‐moving flame fronts.