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The differential Riccati equation appears in different fields of appliedmathematics like control and system theory. Recently Galerkin methods based onKrylov subspaces were developed for the autonomous differential Riccatiequation. These methods overcome the prohibitively large storage requirementsand computational costs of the numerical solution. In view of memory efficientapproximation, we review and extend known solution formulas and identifyinvariant subspaces for a possibly low-dimensional solution representation.Based on these theoretical findings, we propose a Galerkin projection onto aspace related to a low-rank approximation of the algebraic Riccati equation.For the numerical implementation, we provide an alternative interpretation ofthe modified \emph{Davison-Maki method} via the transformed flow of thedifferential Riccati equation, which enables us to rule out known stabilityissues of the method in combination with the proposed projection scheme. Wepresent numerical experiments for large-scale autonomous differential Riccatiequations and compare our approach with high-order splitting schemes.

Invariant Galerkin Ansatz Spaces and Davison-Maki Methods for the Numerical Solution of Differential Riccati Equations