  The efficient numerical integration of large-scale matrix differential
equations is a topical problem in numerical analysis and of great importance in
many applications. Standard numerical methods applied to such problems require
an unduly amount of computing time and memory, in general. Based on a dynamical
low-rank approximation of the solution, a new splitting integrator is proposed
for a quite general class of stiff matrix differential equations. This class
comprises differential Lyapunov and differential Riccati equations that arise
from spatial discretizations of partial differential equations. The proposed
integrator handles stiffness in an efficient way, and it preserves the symmetry
and positive semidefiniteness of solutions of differential Lyapunov equations.
Numerical examples that illustrate the benefits of this new method are given.
In particular, numerical results for the efficient simulation of the weather
phenomenon El Ni\~no are presented.
