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<p>In this work, we introduce high-order Basis-Update &amp; Galerkin (BUG) integrators based on explicit Runge–Kutta methods for large-scale matrix differential equations. These dynamical low-rank integrators extend the BUG integrator [<CitationRef CitationID="CR1">1</CitationRef>] 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.</p>

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High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge–Kutta Methods

  • 期刊论文

  • 发表日期：

    2026年5月13日

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  • Fabio Nobile,
  • Sébastien Riffaud

摘要

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 [1] 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.