The Hopf algebra of Lie group integrators has been introduced by H. Munthe-Kaas and W. Wright as a tool to handle Runge-Kutta numerical methods on homogeneous spaces. It is spanned by planar rooted forests, possibly decorated. We will describe a canonical surjective Hopf algebra morphism onto the shuffle Hopf algebra which deserves to be called planar arborification. The space of primitive elements is a free post-Lie algebra, which in turn will permit us to describe the corresponding co-arborification process.
Joint work with Charles Curry (NTNU Trondheim), Kurusch Ebrahimi-Fard (NTNU) and Hans Z. Munthe-Kaas (Univ. Bergen).
The two triangles appearing at 24'04" and 25'19'' respectively should be understood as a #.[-]

When designing high order schemes for solving time-dependent kinetic and related PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes. In this talk we discuss two classes of high order time discretization, i.e, the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performance of these schemes.[-]