  We consider a Krylov subspace approximation method for the symmetric
differential Riccati equation $\dot{X} = AX + XA^T + Q - XSX$, $X(0)=X_0$. The
method we consider is based on projecting the large scale equation onto a
Krylov subspace spanned by the matrix $A$ and the low rank factors of $X_0$ and
$Q$. We prove that the method is structure preserving in the sense that it
preserves two important properties of the exact flow, namely the positivity of
the exact flow, and also the property of monotonicity. We also provide a
theoretical a priori error analysis which shows a superlinear convergence of
the method. This behavior is illustrated in the numerical experiments.
Moreover, we derive an efficient a posteriori error estimate as well as discuss
multiple time stepping combined with a cut of the rank of the numerical
solution.
