In this paper, we consider a modified Douglas splitting method for a class of differential matrix equations, including differential Lyapunov and differential Riccati equations. The method we consider is based on a natural three-term splitting of the equations. The implementation of the algorithm requires only the solution of a linear algebraic system with multiple right-hand sides in each time step. It is proved that the method is convergent of order two and it preserves the symmetry and positive semidefiniteness of solutions of differential Lyapunov equations. Moreover, we show how the method can be handled in a low-rank setting for large-scale computations. We also provide a theoretical a priori error analysis for the low-rank algorithms. Numerical results are presented to verify the theoretical analysis.

本文针对一类微分矩阵方程，包括微分Lyapunov方程和微分Riccati方程，考虑了一种改进的Douglas分裂方法。我们考虑的方法基于方程的自然三项分解。该算法的实现仅需要在每个时间步中具有多个右侧的线性代数系统的解。证明了该方法是二阶收敛的，并且保留了微分Lyapunov方程解的对称性和正半定性。此外，我们展示了如何在低秩设置中处理该方法以进行大规模计算。我们还为低秩算法提供了理论上的先验误差分析。数值结果表明了理论分析的正确性。