Mechthild Thalhammer, Universität Innsbruck (Koautor: Alexander Ostermann)
We consider fully nonlinear initial-boundary value problems of parabolic type modelling for example nonlinear diffusion or heat conduction processes. When we apply a numerical method for discretizing such a problem in time, a matter of interest is whether the discretization reproduces the trajectories and whether it captures the dynamics of the underlying problem correctly.
For the analysis, we write the partial differential equation as abstract ordinary differential equation, and we work in the framework of analytic semigroups and Banach space valued Hölder continuous functions. For the classes of implicit Runge-Kutta and linear multistep methods we show that the quantitative and qualitative properties of the original problem are well preserved.