Invited Lecture
Stability and efficiency enhancements of operator-splitting methods
University of Saskatchewan, Canada
Abstract
Operator-splitting methods are widely used for the time integration of differential equations, especially those that arise from multi-scale or multi-physics models, because a monolithic 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 impacted by instability. However, besides the methods themselves, there are many other ancillary aspects to an overall operator-splitting method that are important in practice but often overlooked. For example, the order in which operators are integrated and the choice of sub-integration methods can significantly affect the performance of an operator-splitting method. In this paper, we design a new four-stage, third-order, 2-split operator-splitting method with seven sub-integrations and an optimized linear stability region. We then propose two general strategies to further improve its stability and efficiency for a specific problem, namely, to choose the ordering of operators to maximize linear stability and to choose low-order explicit sub-integrators for unstable sub-integrations. We demonstrate about a 40% improvement in the performance from the combined use of these strategies relative to standard implementations on a benchmark problem from cardiac electrophysiology.
Joint work with Siqi Wei and Victoria Guenter, University of Saskatchewan