High order methods for elliptic and time dependent reaction–diffusion singularly perturbed problems C. Clavero * , J.L. Gracia Department of Applied Mathematics, University of Zaragoza, C Maria de Luna 3, Zaragoza 50018, Spain Abstract The objective of this paper is to construct some high order uniform numerical meth- ods to solve linear reaction–diffusion singularly perturbed problems. First, for 1D ellip- tic problems, based on the central finite difference scheme, a new HODIE method is defined on a piecewise uniform Shishkin mesh. Using this HODIE scheme jointly with a two stage SDIRK method, we solve a 1D parabolic singularly perturbed problem. In both cases we prove that the methods are third-order uniform convergent in the maxi- mum norm. Finally, for a 2D parabolic problem of the same type, we show numerically that the combination of the HODIE scheme with a fractional step RK method gives again a third-order uniform convergent scheme. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Reaction–diffusion problems; HODIE schemes; SDIRK method; Fractional RK method; Uniform convergence; High order 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.10.007 * Corresponding author. E-mail addresses: clavero@unizar.es (C. Clavero), jlgracia@unizar.es (J.L. Gracia). Applied Mathematics and Computation 168 (2005) 1109–1127 www.elsevier.com/locate/amc