[Paper Review] Relaxed Multirate Infinitesimal Step Methods for Initial-Value Problems

Jean Sexton, Daniel R. Reynolds|arXiv (Cornell University)|Aug 10, 2018

Numerical methods for differential equations44 references4 citations

TL;DR

This paper introduces a new class of fourth-order accurate multirate time integration methods—Relaxed Multirate Infinitesimal Step (RMIS) methods—by extending the Recursive Flux Splitting Multirate (RFSMR) framework using Generalized Additive Runge-Kutta (GARK) theory. The method enables subcycling, adaptive time-scale separation, and embedded error estimation without sacrificing linear stability, achieving high-order accuracy in multirate ODE systems with distinct fast and slow dynamics.

ABSTRACT

This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate (RFSMR) version of the Multirate Infinitesimal Step (MIS) methods and use recent theoretical developments for Generalized Additive Runge-Kutta methods to propose our higher-order Relaxed Multirate Infinitesimal Step extensions. The resulting framework supports a range of attractive properties for multirate methods, including telescopic extensions, subcycling, embeddings for temporal error estimation, and support for changes to the fast/slow time-scale separation between steps, without requiring any sacrifices in linear stability. In addition to providing rigorous theoretical developments for these new methods, we provide numerical tests demonstrating convergence and efficiency on a suite of multirate test problems.

Motivation & Objective

• Address the need for high-order, stable, and efficient time integrators in multiphysics simulations with widely varying time scales.
• Overcome limitations of existing multirate methods that sacrifice stability or accuracy when handling variable time-scale separations.
• Develop a framework that supports both explicit and implicit integration of fast components while maintaining fourth-order accuracy.
• Enable practical features such as temporal error estimation, telescopic extensions, and dynamic adjustment of time-scale separation during integration.
• Ensure linear stability is preserved even when the fast and slow time-scale ratio changes between steps.

Proposed method

• Formulate the multirate problem as an additive split ODE: y' = f^f(y) + f^s(y), with distinct fast and slow components.
• Apply GARK (Generalized Additive Runge-Kutta) theory to derive order conditions and stability properties for the multirate scheme.
• Construct the RMIS method by relaxing the strict coupling constraints of standard MIS methods, allowing for higher-order accuracy.
• Use a two-stage structure: an outer Runge-Kutta method for the slow component and an inner method for the fast component, with embedded stages for error estimation.
• Integrate the fast component using m substeps per slow step, with time-step ratio m ≈ dt_s / dt_f, enabling subcycling.
• Ensure linear stability by preserving the stability properties of the underlying base methods through careful coefficient design.

Experimental results

Research questions

• RQ1 Can a fourth-order accurate multirate method be constructed that maintains linear stability under variable time-scale separation?
• RQ2 How can subcycling and temporal error estimation be incorporated into multirate schemes without degrading stability or accuracy?
• RQ3 What is the role of GARK theory in enabling higher-order multirate methods with relaxed coupling constraints?
• RQ4 Can the method support dynamic changes in the fast/slow time-scale ratio during integration while preserving convergence and stability?
• RQ5 What are the practical efficiency gains of the new method compared to standard multirate or single-rate integrators on multirate ODE problems?

Key findings

• The proposed Relaxed Multirate Infinitesimal Step (RMIS) methods achieve fourth-order accuracy for multirate ODE systems with distinct fast and slow components.
• The method supports subcycling, telescopic extensions, and embedded error estimation for adaptive time stepping, all without compromising linear stability.
• The framework allows for dynamic adjustment of the time-scale separation ratio between steps, enabling greater flexibility in complex simulations.
• Numerical tests confirm optimal convergence rates and demonstrate significant efficiency improvements over single-rate methods on multirate test problems.
• Theoretical analysis confirms that the method satisfies all required order conditions and linear stability constraints, even under relaxed coupling assumptions.
• The method is applicable to both explicit and implicit integration of the fast component, broadening its applicability to stiff and non-stiff systems.