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arXiv · 2502.07040

Robust high-order low-rank BUG integrators based on explicit Runge--Kutta methods

Fabio Nobile·Sébastien Riffaud
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Abstract

In this work, we introduce high-order Basis-Update & Galerkin (BUG) integrators based on explicit Runge-Kutta methods for large-scale matrix differential equations. These dynamical low-rank integrators extend the BUG integrator to arbitrary explicit Runge-Kutta schemes by performing a BUG step at each stage of the method. The resulting Runge-Kutta BUG (RK-BUG) integrators are robust with respect to small singular values, fully forward in time, and high-order accurate, while enabling conservation and rank adaptivity. We prove that RK-BUG integrators retain the order of convergence of the underlying Runge-Kutta method until the error reaches a plateau corresponding to the low-rank truncation error, which vanishes as the rank becomes full. This theoretical analysis is supported by several numerical experiments. The results demonstrate the high-order convergence of the RK-BUG integrator and its superior accuracy compared to other existing dynamical low-rank integrators.

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Fabio Nobile, Sébastien Riffaud. 2025-02-10. Robust high-order low-rank BUG integrators based on explicit Runge--Kutta methods. https://arxiv.org/abs/2502.07040

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