Adaptive low-rank exponential integrators for large-scale differential Riccati equation

TL;DR

This paper introduces adaptive low-rank exponential integrators for large-scale differential Riccati equations, improving accuracy and efficiency during transient and steady-state phases through embedded schemes and adaptive step-size control.

Contribution

It presents a novel combination of embedded exponential Rosenbrock schemes with adaptive step-size control for large-scale stiff DREs.

Findings

Enhanced accuracy during transient phases

Improved computational efficiency over fixed-step methods

Consistent performance on benchmark problems

Abstract

Matrix differential Riccati equation (DRE) typically exhibits transient and steady-state phases, posing challenges for fixed-step time integration methods, which may lack accuracy during transients or oversample in steady regimes. In this work, we propose adaptive low-rank matrix-valued exponential integrators for large-scale stiff DRE. The methods combine embedded exponential Rosenbrock-type schemes and adaptive step-size control, enabling an automatic adjustment to the evolving solution dynamics. This improves the accuracy during rapid transient phases while maintaining high accuracy in the steady state. Numerical experiments on benchmark problems demonstrate that the proposed adaptive integrators consistently improve accuracy and computational efficiency compared with fixed-step low-rank schemes.