A note on implementations of the Boosting Algorithm and Heterogeneous Multiscale Methods

TL;DR

This paper improves convergence understanding of the Boosting Algorithm and clarifies the accuracy limitations of Heterogeneous Multiscale Methods when applied to stiff ODEs, highlighting their theoretical properties.

Contribution

It provides new convergence results for BA and shows that HMM's accuracy is limited to first order in macro time steps, regardless of solver order.

Findings

Enhanced convergence results for the Boosting Algorithm.

HMM's accuracy is first order in macro time step, independent of solver order.

Unified formulation of HMM for dissipative stiff ODEs.

Abstract

We present improved convergence results for the Boosting Algorithm (BA), and demonstrate that an existing formulation of the Heterogeneous Multiscale Methods (HMM) is accurate to first order only in the macro time step, regardless of the order of the numerical solvers employed. These results are obtained by considering the BA and two other formulations of HMM as special cases of a general formulation of HMM applied to dissipative stiff ordinary differential equations.