Efficient and Flexible Multirate Temporal Adaptivity

TL;DR

This work tackles efficient adaptive time stepping for multirate infinitesimal MRI methods solving ODEs with multiple time scales. It introduces two multirate controller families, Decoupled and $H$-$Tol$, and demonstrates their superior robustness and efficiency across benchmark problems when paired with embedded MRI methods, including new MERK embeddings up to order 5. Through extensive numerical experiments and a nested three-scale test, the authors show that these controllers achieve high accuracy at reduced cost, outperforming prior $H$-$h$ approaches in strongly multiscale settings. The results provide practical guidance for selecting MRI methods and controllers in real-world multiscale simulations and extend the toolkit with a fifth-order embedded MRI method.

Abstract

In this work we present two new families of multirate time step adaptivity controllers, that are designed to work with embedded multirate infinitesimal (MRI) time integration methods for adapting time steps when solving problems with multiple time scales. We compare these controllers against competing approaches on two benchmark problems and see that they offer dramatically improved performance and flexibility, with each proposed family excelling on different types of multirate applications. The combination of embedded MRI methods and the proposed controllers enable adaptive simulations of problems with a potentially arbitrary number of time scales, achieving high accuracy while maintaining low computational cost. Additionally, we introduce a new set of embeddings for the family of explicit multirate exponential Runge--Kutta (MERK) methods of orders 2 through 5, resulting in the first-ever fifth-order embedded MRI method. Finally, we compare the performance of a wide range of embedded MRI methods on our benchmark problems to provide guidance on how to select an appropriate MRI method and multirate controller.

Figures (5)

• Figure 1: Accuracy measurements for each multirate controller family, when tested across a wide range of tolerances and individual MRI controllers. Columns denote MRI method accuracies: left are $\mathcal{O}(H^2)$, middle are $\mathcal{O}(H^3)$, and right are $\mathcal{O}(H^4)$ and $\mathcal{O}(H^5)$. The KPR test problem is in the top two rows, and the Brusselator test problem is in the bottom two rows. Note that where two regions overlap, their shading mixes together; the $H\text{-}Tol$ and Decoupled families perfectly overlap, resulting in a teal-gray color.
• Figure 2: Efficiency comparisons for the top second-order adaptive MRI methods. The top row contains the slow and fast time scales for the KPR test problem with multirate ratios $\omega=\{50,500\}$. The stiff Brusselator test problem with both stiffness parameters $\epsilon = \{10^{-4},10^{-5}\}$ is on the bottom.
• Figure 3: Efficiency comparisons for the top third-order adaptive MRI methods. The top row shows the slow and fast time scales for the KPR test problem with multirate ratios $\omega=\{50,500\}$. The stiff Brusselator test problem with both stiffness parameters $\epsilon = \{10^{-4},10^{-5}\}$ is on the bottom row.
• Figure 4: Efficiency comparisons for the top fourth- and fifth-order adaptive MRI methods. The top row shows the slow and fast time scales for the KPR test problem with multirate ratios $\omega=\{50,500\}$. The stiff Brusselator test problem with both stiffness parameters $\epsilon = \{10^{-4},10^{-5}\}$ is on the bottom row.
• Figure 5: Slow and fast time step histories for each multirate controller family on the KPR and stiff Brusselator test problems. The legends list the numbers of slow and fast time steps, as well as the attained solution accuracy factor \ref{['eq:accuracy']}.