Second or fourth-order finite difference operators, which one is most effective? IJSM Second or fourth-order finite difference operators, which one is most effective? Kolade M. Owolabi Department of Mathematical Sciences, Federal University of Technology, Akure, P.O. Box 704, Ondo State, Nigeria. Email: mkowolax@yahoo.com This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined. 2010 Mathematics Subject Classification: {92B05, 92C15, 35K55, 65M20, 76R50} Keywords: Consistency, Finite difference, PDEs, Reaction-diffusion, Stability, Symmetry. INTRODUCTION Reaction-diffusion equations are classified as a special class of parabolic time-dependent partial differential equations. The major way of solving the class of these equations is through discretization. A well-known approach to solve time-dependent partial differential equation, with solutions varying in both time and space, is the method of lines (MOL), see Ascher et al. (1995), Schisser and Griffits (2009), Holden and Karlsen (2012) and Strikwerda (2004) for details. Application of this method requires to first constructing a semi-discrete approximation to the problem by setting up a regular grid in space, this is achieved by discretising the spatial independent variables with boundary constraints. Hence, a couple systems of ordinary differential equations are generated in time, which is associated with the initial value. Once that is done, we numerically approximate the solutions to the original time- dependent partial differential equation by marching forward in time on this grid. Conveniently, we can now apply any existing, and generally well established, time-stepping numerical methods such as the implicit-explicit (IMEX) schemes, Runge-Kutta methods or exponential time differencing (ETD) schemes among many others. In this paper, for the spatial discretisation, we are primarily concerned with the use of higher-order finite difference method. The discrete approximation to the derivatives will be converted into Toeplitz matrices. The discretisation in time uses majorly the exponential time differencing schemes, other time-stepping methods include but not limited to the fourth-order Runge-Kutta (RK4) method and implicit-explicit schemes. A brief of each of these methods will be International Journal of Statistics and Mathematics Vol. 1(3), pp. 044-054, August, 2014. © www.premierpublishers.org, ISSN: 2375-0499x Review