[Paper Review] A self adjusting multirate algorithm based on the TR-BDF2 method

Luca Bonaventura, Francesco Casella+2 more|arXiv (Cornell University)|Jan 27, 2018

Numerical methods for differential equations13 references3 citations

TL;DR

This paper proposes a self-adjusting multirate time integration method based on the TR-BDF2 solver for stiff ODEs and hyperbolic PDEs. By dynamically adapting time steps per component using error estimators and leveraging TR-BDF2's embedded error control and cubic Hermite interpolation, the method achieves significant computational efficiency gains—especially for problems with localized dynamics like shock waves—while maintaining accuracy and stability across stiff and hyperbolic problems.

ABSTRACT

We propose a self adjusting multirate method based on the TR-BDF2 solver. The potential advantages of using TR-BDF2 as the key component of a multirate framework are highlighted. A linear stability analysis of the resulting approach is presented and the stability features of the resulting algorithm are analysed. The analysis framework is completely general and allows to study along the same lines the stability of self adjusting multirate methods based on a generic one step solver. A number of numerical experiments demonstrate the efficiency and accuracy of the resulting approach also the time discretization of hyperbolic partial differential equations.

Motivation & Objective

• Address the computational inefficiency of single-rate time integration in systems with widely varying time scales.
• Overcome stability and efficiency limitations in existing multirate methods, especially for stiff and hyperbolic problems.
• Leverage the TR-BDF2 method’s favorable properties—L-stability, embedded error estimation, and continuous output—for robust multirate integration.
• Enable automatic detection of active (fast) and latent (slow) components through adaptive time step refinement.
• Demonstrate the method’s effectiveness on benchmark problems, including nonlinear hyperbolic PDEs with shock waves and rarefaction waves.

Proposed method

• Adopt a self-adjusting multirate strategy where a global time step is tentatively taken using TR-BDF2, then refined only for components exceeding local error tolerance.
• Use the TR-BDF2 method as the base single-rate integrator due to its second-order accuracy, L-stability, and embedded third-order error estimator.
• Apply a partitioning criterion based on local error estimation to classify components as active (requiring small steps) or latent (using larger steps).
• Employ cubic Hermite interpolation for continuous output to maintain accuracy during time step refinement, especially critical for stiff and transient phenomena.
• Implement a recursive time stepping strategy that avoids global stiffness constraints by localizing refinement only where needed.
• Use a linear stability analysis framework applicable to any one-step solver, applied here to TR-BDF2 in a multirate context.

Experimental results

Research questions

• RQ1 Can a self-adjusting multirate method based on TR-BDF2 achieve improved stability and efficiency for stiff and hyperbolic problems compared to single-rate solvers?
• RQ2 How does the stability of the multirate TR-BDF2 method behave under varying coupling strength and stiffness, particularly for problems with purely imaginary eigenvalues?
• RQ3 To what extent does the use of cubic Hermite interpolation improve accuracy and reduce step rejection in multirate schemes compared to linear interpolation?
• RQ4 Can the method automatically detect and resolve localized phenomena such as shock waves and rarefaction waves in hyperbolic PDEs with minimal user intervention?
• RQ5 What are the computational savings and accuracy trade-offs when applying the method to benchmark nonlinear hyperbolic PDEs like the Burgers equation?

Key findings

• The self-adjusting multirate TR-BDF2 method achieves significant computational efficiency gains—especially in problems with localized dynamics like shock waves—by reducing the number of time steps for latent components.
• Numerical experiments show that cubic Hermite interpolation is essential for performance: linear interpolation led to excessive step rejections and no efficiency gain over single-rate TR-BDF2.
• Relative errors from the multirate TR-BDF2 with cubic Hermite interpolation, linear interpolation, and single-rate TR-BDF2 were comparable, indicating consistent accuracy across methods.
• When compared to a high-accuracy reference solution (using MATLAB's ode45), all variants showed errors reduced by several orders of magnitude, confirming that spatial and temporal discretization errors were well-controlled.
• The method demonstrated good stability properties even for problems with purely imaginary eigenvalues—indicating potential utility for hyperbolic PDEs and structural dynamics—despite no complete theoretical stability proof.
• For the rarefaction wave case, the fraction of active components was higher and computational savings were less pronounced than in the shock wave case, reflecting the more widespread activity in the solution.