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 lowrank 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 Niño are presented.

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We consider the numerical approximation of stochastic partial differential equations (SPDEs) based models for a quasi-periodic climate pattern in the tropical Pacific Ocean known as El Niño phenomenon. We show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation, respectively. In this context we provide a numerical framework to approximate these parameters directly. We compare this method to stochastic differential equations and SPDEs based models from the literature solved by Taylor methods and stochastic Galerkin methods, respectively. Numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 (last Niño event) validate the efficiency of our approach.

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