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

Initial boundary value problems (IBVPs) u'(t) = Au(t) + f(t), ∂u(t) = g(t), 0 ≤ t ≤ T, u(0) = u0, where A: D(A) ⊂ X → X and ∂: D(A) ⊂ 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.