The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently, Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. Known solution formulas are reviewed and extended. Because of memory-efficient approximations, invariant subspaces for a possibly low-dimensional solution representation are identified. A Galerkin projection onto a trial space related to a low-rank approximation of the solution of the algebraic Riccati equation is proposed. The modified Davison-Maki method is used for time discretization. Known stability issues of the Davison-Maki method are discussed. Numerical experiments for large-scale autonomous differential Riccati equations and a comparison with high-order splitting schemes are presented.

微分 Riccati 方程出现在应用数学的不同领域，如控制和系统论。最近，基于 Krylov 子空间的 Galerkin 方法被开发用于自治微分 Riccati 方程。这些方法克服了数值解的巨大存储要求和计算成本。已知的解决方案公式被审查和扩展。由于内存有效的近似值，识别出可能低维解表示的不变子空间。提出了对与代数 Riccati 方程解的低秩近似相关的试验空间的伽辽金投影。所述改性的Davison-希方法用于时间离散。Davison-Maki 方法的已知稳定性问题进行了讨论。给出了大规模自主微分 Riccati 方程的数值实验以及与高阶分裂方案的比较。