Optimal orders of convergence for Runge-Kutta methods and linear, initial boundary value problems

Initial boundary value problems (IBVPs) u'(t) = <(A)overtilde>u (t) + f (t), partial derivativeu(t) = g(t), 0 less than or equal to t less than or equal to T, u(0) = u(0), where (A) over tilde :D((A) over tilde) subset of X --> X and partial derivative:D((A) over tilde) subset of X --> Y are linear, densely defined operators and X and Y are Banach spaces, are considered. These IBVPs are discretized by means of a Runge-Kutta method. Optimal error estimates are provided. Examples of applications to hyperbolic and parabolic problems are given and some numerical experiments are presented. These experiments confirm that the error estimates are of optimal order. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.