Improving the stability and efficiency of high-order operator-splitting methods

TL;DR

This paper develops new high-order operator-splitting methods with enhanced stability and efficiency for differential equations, especially in multi-physics models, by optimizing operator ordering, sub-integration choices, and stability criteria.

Contribution

It introduces a novel four-stage, third-order, 2-split operator-splitting method with optimized stability and a general principle for improving stability using explicit sub-integrators.

Findings

Proposed a new third-order, 2-split operator-splitting method with seven sub-integrations.

Achieved nearly 30% performance improvement over existing methods.

Demonstrated effectiveness on a cardiac electrophysiology benchmark problem.

Abstract

Operator-splitting methods are widely used to solve differential equations, especially those that arise from multi-scale or multi-physics models, because a monolithic (single-method) approach may be inefficient or even infeasible. The most common operator-splitting methods are the first-order Lie--Trotter (or Godunov) and the second-order Strang (Strang--Marchuk) splitting methods. High-order splitting methods with real coefficients require backward-in-time integration in each operator and hence may be adversely impacted by instability for certain operators such as diffusion. However, besides the method coefficients, there are many other ancillary aspects to an overall operator-splitting method that are important but often overlooked. For example, the operator ordering and the choice of sub-integration methods can significantly affect the stability and efficiency of an operator-splitting method. In this paper, we investigate some design principles for the construction of operator-splitting methods, including minimization of local error measure, choice of sub-integration method, maximization of linear stability, and minimization of overall computational cost. We propose a new four-stage, third-order, 2-split operator-splitting method with seven sub-integrations per step and optimized linear stability for a benchmark problem from cardiac electrophysiology. We then propose a general principle to further improve stability and efficiency of such operator-splitting methods by using low-order, explicit sub-integrators for unstable sub-integrations. We demonstrate an almost 30\% improvement in the performance of methods derived from these design principles compared to the best-known third-order methods.