Avoiding Order Reduction of Runge–Kutta Discretizations for Linear Time-Dependent Parabolic Problems

Abstract

A technique is developed in this paper to avoid order reduction when discretizing linear parabolic problems with time dependent operator using Runge–Kutta methods in time and standard schemes in space. In an abstract framework, the boundaries of the stages of the Runge–Kutta method which would completely avoid the order reduction are given. Then, the possible practical implementations for the calculus of those boundaries from the given data are studied, and the full discretization is completely analyzed. Some numerical experiments are included.