Parallelization of Implicit-Explicit Runge-Kutta Methods for Cluster of PCs Jos´ e Miguel Mantas 1 , Pedro Gonz´ alez 2 , and Jos´ e A. Carrillo 3 1 Software Engineering Department. University of Granada C/ P. Daniel de Saucedo s/n. E-18071 Granada, Spain jmmantas@ugr.es 2 Department of Applied Mathematics. University of Granada Avda. Fuentenueva s/n. E-18071 Granada, Spain prodelas@ugr.es 3 Departament de Matem`atiques - ICREA. Universitat Aut`onoma de Barcelona Bellaterra E-08193 carrillo@mat.uab.es Abstract. Several physical phenomena of great importance in science and engineering are described by large partly stiff differential systems where the stiff terms can be easily separated from the remaining terms. Implicit-Explicit Runge-Kutta (IMEXRK) methods have proven to be useful solving these systems efficiently. However, the application of these methods still requires a large computational effort and their parallel im- plementation constitutes a suitable way to achieve acceptable response times. In this paper, a technique to parallelize and implement efficiently IMEXRK methods on PC clusters is proposed. This technique has been used to parallelize a particular IMEXRK method and an efficient parallel implementation of the resultant scheme has been derived in a structured manner by following a component-based approach. Several numerical experiments which have been performed on a cluster of dual PCs reveal the good speedup and the satisfactory scalability of the parallel solver obtained. 1 Introduction The spatial discretization of a great variety of time-dependent partial differential equations (PDEs) by the method of lines leads to large systems of ordinary differential equations (ODEs) with this form: dy dt = f (y)+ g(y), y(0) = y 0 ∈ IR d , t> 0 (1) where y = y(t) ∈ IR d is the unknown function of a d-dimensional ODE system which is defined by the component functions f , g : IR d -→ IR d . The function g(y) results from the discretization of the stiff terms and f (y) results from the discretization of the remaining terms. The function g is usually written as (1/ǫ)˜ g (˜ g : IR d -→ IR d ), where ǫ> 0 is the stiffness parameter [5]. J.C. Cunha and P.D. Medeiros (Eds.): Euro-Par 2005, LNCS 3648, pp. 815–825, 2005. c  Springer-Verlag Berlin Heidelberg 2005