  We consider high-order splitting schemes for large-scale differential Riccati
equations. Such equations arise in many different areas and are especially
important within the field of optimal control. In the large-scale case, it is
critical to employ structural properties of the matrix-valued solution, or the
computational cost and storage requirements become infeasible. Our main
contribution is therefore to formulate these high-order splitting schemes in a
efficient way by utilizing a low-rank factorization. Previous results indicated
that this was impossible for methods of order higher than 2, but our new
approach overcomes these difficulties. In addition, we demonstrate that the
proposed methods contain natural embedded error estimates. These may be used
e.g. for time step adaptivity, and our numerical experiments in this direction
show promising results.
