Numerical low-rank approximation of matrix differential equations
This work addresses the need for efficient numerical integration of large-scale matrix differential equations, which is important for applications like weather simulation.
The paper proposes a new splitting integrator for stiff matrix differential equations using dynamical low-rank approximation, which efficiently handles stiffness and preserves symmetry and positive semidefiniteness. Numerical examples, including simulation of El Niño, demonstrate the method's benefits.
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 Niño are presented.