Robust High-Order Projector-Splitting Integrators

Abstract

We develop a general framework for constructing robust high-order projector-splitting integrators for dynamical low-rank approximation. For a prescribed matrix increment, we show that the standard K-S-L projector-splitting step is equivalent to a reduced K-L step, thereby eliminating the explicit backward S-step. We then establish a central relaxed exactness property of the standard projector-splitting integrator: the rank-$r$ approximation inherits the accuracy of the numerical matrix increment, with an error bound independent of small singular values. The resulting schemes evolve fixed rank-$r$ factors and require neither basis augmentation nor rank truncation. As concrete examples, we combine the framework with selected second- and third-order Runge--Kutta methods to obtain robust high-order projector-splitting integrators. Numerical experiments confirm the predicted uniform convergence rates with respect to small singular values.