Apdative timestep control for fully implicit Runge-Kutta methods of higher order

It is possible to construct fully implicit Runge-Kutta methods like Gauß-Legendre, Radau-IA, Radau-IIA, Lobatto-IIIA, -IIIB, and -IIIC methods of arbitrary high order of convergence. The aim of this paper is to find a new adaptive time stepping for these classes which is based on the embedding technique. Adaptive time step control with embedding is well-known for Runge-Kutta methods, and therefore new embedded methods of order s-1 for the above classes of fully implicit Runge-Kutta methods are constructed. Since these fully implicit methods need the solution of a huge non-linear system of equations different approaches for non-linear equations are discussed and compared. It can be observed that non-linear solvers like the usually used simplified Newton method have a step size restriction if they are applied on higher order methods. We apply our new methods on some lower dimensional ODEs to show that our approach leads to an efficient method.