Cambridge International Science Publishing Cambridge CB1 6AZ Great Britain Journal of Computational Methods in Sciences and Engineering vol. 3, no. 3, 2003, pp. 403-413 ISSN 1472–7978 High Order Uniformly Convergent Fractional Step RK Methods and HODIE Finite Difference Schemes for 2D Evolutionary Convection-Diffusion Problems J.L. Gracia 1 Departamento de Matem´atica Aplicada, Universidad de Zaragoza, Zaragoza, Spain J.C. Jorge 2 Departamento de Matem´atica e Inform´ atica, Universidad P´ ublica de Navarra, Pamplona, Spain B. Bujanda 3 Departamento de Matem´aticas y Computaci´on, Universidad de La Rioja, Logro˜ no, Spain C. Clavero 45 Departamento de Matem´atica Aplicada, Centro Polit´ ecnico, C/Mar´ ıa de Luna, 3, 50018 Zaragoza, Spain Received 15 October, 2002; accepted in revised form 10 January, 2003 Abstract: In this work we present a numerical method to solve linear time dependent two dimensional singularly perturbed problems of convection-diffusion type with dominat- ing convection term; this class of problems is characterized by the presence of a regular boundary layer in the output boundary of the spatial domain. The method combines the alternating direction technique, based on an A-stable third order RK method, with a third order HODIE finite difference scheme of classical type, i.e., exact only on polynomial functions, constructed on a special spatial mesh of Shishkin type. We show that, under appropriate restrictions between the discretization parameters, the method is uniformly convergent with respect to the diffusion parameter, having order three (except by a loga- rithmic factor) in the maximum norm. The method provides the computational advantages of the splitting technique and also the efficiency provided by high order methods, achieving good approximations of the solution in the whole domain, including the boundary layer region. We show some numerical results validating in practice the good properties of the method. Keywords: Fractional steps, HODIE methods, high order, uniform convergence. Mathematics Subject Classification: 65N12, 65N30, 65N06. 1 E-Mail: jlgracia@posta.unizar.es 2 E-Mail: jcjorge@unavarra.es 3 E-Mail: bbujanda@dmc.unirioja.es 4 Corresponding author 5 Phone: +34 976 761988 Fax: +34 976 761886 E-Mail: clavero@posta.unizar.es