Applied Numerical Mathematics 62 (2012) 1463–1476 Contents lists available at SciVerse ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Variable step-size fractional step Runge–Kutta methods for time-dependent partial differential equations L. Portero ∗ , A. Arrarás, J.C. Jorge Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, Edificio de Las Encinas, Campus de Arrosadía, 31006 Pamplona, Spain article info abstract Article history: Available online 7 June 2012 Keywords: Alternating direction implicit Domain decomposition Fractional step Runge–Kutta method Parabolic problem Variable step-size Fractional step Runge–Kutta methods are a class of additive Runge–Kutta schemes that provide efficient time discretizations for evolutionary partial differential equations. This efficiency is due to appropriate decompositions of the elliptic operator involving the spatial derivatives. In this work, we tackle the design and analysis of embedded pairs of fractional step Runge–Kutta methods. Such methods suitably estimate the local error at each time step, thus providing efficient variable step-size time integrations. Finally, some numerical experiments illustrate the behaviour of the proposed algorithms.  2012 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction The numerical solution of time-dependent partial differential equations (PDEs) is usually achieved by a proper combina- tion of suitable spatial discretization and time integration processes. In this paper, we are mainly concerned with the study of the latter process in the context of linear parabolic problems. Plenty of methods have been proposed along the years for the time integration of such problems. Classical schemes, such as Runge–Kutta (RK) methods, have the drawback of a high computational cost, especially when dealing with multi-dimensional equations. The so-called splitting methods provide an efficient alternative to these schemes if a suitable partitioning of the time derivative function is performed. Within the class of splitting methods, we shall focus our attention on m-part additive Runge–Kutta (ARK m ) schemes. These methods were introduced by Cooper and Sayfy in [8,9] and subsequently considered by many other authors (cf. [2,6, 13] and references therein). Basically, they are one-step time integrators that merge m different RK schemes into a single composite method, thus permitting to solve evolutionary PDEs whose time derivative has been previously decomposed into m terms. Some celebrated time integrators formally belong to the set of ARK m methods. That is the case, for instance, of implicit– explicit (IMEX) schemes (cf. [4]). More precisely, an IMEX scheme can be seen as an ARK 2 scheme which combines both implicit and explicit RK methods (for handling stiff and non-stiff terms, respectively). In the context of time-dependent reaction–diffusion problems, IMEX schemes have been successfully applied to spatially discretized equations involving a linear diffusion term (responsible for the stiffness of the system) and a nonlinear reaction term (cf. [19]). If we consider an arbitrary number m of parts, there exists a special family of ARK m methods known as fractional step Runge–Kutta (FSRK m ) methods. These time integrators, which typically comprise m diagonally implicit RK schemes, have been analyzed by [5,6] within the framework of linear parabolic problems. FSRK m methods may be combined with various partitioning techniques for the time derivative function: if the splitting is based on different spatial variables, it is referred to as a dimensional or component-wise splitting; otherwise, if it is related to a suitable decomposition of the spatial domain, it is called a domain decomposition splitting. The first type of splitting reformulates the original multi-dimensional problem * Corresponding author. Tel.: +34 948168052; fax: +34 948169521. E-mail address: laura.portero@unavarra.es (L. Portero). 0168-9274/$36.00  2012 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apnum.2012.06.015