Variational multirate integrators

TL;DR

The paper develops variational multirate integrators for systems with slow and fast dynamics by extending discrete variational mechanics to a two-grid (macro/micro) discretization. The resulting integrators are symplectic and momentum-preserving, with convergence characterized by the rule $r = \min(2l,u)$ (where $l$ is the polynomial degree and $u$ the quadrature order), and stability analyzed for key split configurations. Numerical experiments on the Fermi–Pasta–Ulam problem and a spring-ring system confirm energy and momentum preservation, verify the predicted convergence orders, and demonstrate meaningful computing-time savings over single-rate methods for practical micro/macro step choices. The work provides a flexible framework that can be extended to optimal control and constrained multirate systems, highlighting the practical impact of structure-preserving multirate methods in multiscale dynamics.

Abstract

The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, a popular approach is to treat the slow and the fast part of a system differently. Embedding this approach in a variational framework is the keystone of this work. By paralleling continuous and discrete variational multirate dynamics, integrators are derived on a time grid consisting of macro and micro time nodes that are symplectic, momentum preserving and also exhibit good energy behaviour. The choice of the discrete approximations for the action determines the convergence order of the scheme as well as its implicit or explicit nature for the different parts of the multirate system. The convergence order is proven using the theory of variational error analysis. The performance of the multirate variational integrators is demonstrated by means of several examples.

Figures (21)

• Figure 3.1: Macro and micro time grid
• Figure 5.1: Linear stability regions depending on the ratio $p = \frac{\omega_s \Delta T}{\omega_s \Delta t}$ for using the trapezoidal rule (left plot) resp. the midpoint rule (right plot) to approximate the action on the micro grid.
• Figure 6.1: Fermi-Pasta-Ulam problem
• Figure 6.2: Spring ring example system with slow and fast configuration
• Figure 6.3: FPU: total energy in the stiff springs and sum of all stiff energies for $p=1$ (left), $p=5$ (centre), and $p=10$ (right) with $\Delta T = 0.3$ for midpoint-midpoint scheme

Theorems & Definitions (20)

• Remark 2.1
• Remark 2.2
• Theorem 3.1
• proof
• Theorem 3.2: Discrete Noether's theorem
• proof
• Theorem 3.3: midpoint fast, midpoint slow
• proof
• Theorem 3.4: midpoint fast, trapezoidal slow
• Theorem 3.5: trapezoidal fast, trapezoidal slow