Publication

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

2026

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 [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.

Official source

https://infoscience.epfl.ch/entities/publication/f2b63a43-1203-4ddd-a07b-96fc307cded3

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/f2b63a43-1203-4ddd-a07b-96fc307cded3

About this result

Related concepts (32)

Runge–Kutta methods

In numerical analysis, the Runge–Kutta methods (ˈrʊŋəˈkʊtɑː ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method".

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Heun's method

In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods.

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

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