Digital Object Identifier (DOI) 10.1007/s002110100332 Numer. Math. (2002) 91: 577–603 Numerische Mathematik Runge-Kutta methods without order reduction for linear initial boundary value problems Isa´ ıas Alonso-Mallo DepartamentodeMatem´ aticaAplicadayComputaci´ on,FacultaddeCiencias,Universidadde Valladolid, c/Doctor Mergelina s/n, 47005 Valladolid, Spain; e-mail: isaias@mac.cie.uva.es Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001 – c  Springer-Verlag 2001 Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretiza- tion of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermedi- ate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Mathematics Subject Classification (1991): 65M20, 65M12, 65M60, 65J10 1. Introduction When applied to stiff systems of ordinary differential equations, the Runge– Kuttamethodssufferfromreductionoforder[13],evenwhenthesolutionis regular. Since the spatial semidiscretization of a partial differential equation becomes stiffer when the spatial discretization is refined, it is natural to observe the order reduction phenomenon when an evolutionary problem in partialdifferentialequationsissolvedbyusingthemethodoflinesapproach. Theorderobservedintheapplicationsisgovernedessentiallybythestage order q of the Runge–Kutta method rather than the classical order p [6,17– 19,21] and it depends on several factors. However, it is well-known that the The author has obtained financial support from DGICYT PB95-705 and JCyL VA025/01