Improving the Stability and Efficiency of Operator-Splitting Methods
Siqi Wei, Victoria Guenter, Raymond J. Spiteri · SIAM Journal on Scientific Computing · 2025
Abstract. Operator-splitting methods are widely used to solve differential equations, especially those that arise from multiscale or multiphysics 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 subintegration 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 subintegration 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 subintegrations 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 subintegrators for unstable subintegrations. We demonstrate an almost 30% improvement in the performance of methods derived from these design principles compared to some well-known high-order methods.