Splitting headache: How well do splitting methods preserve stability?

Abstract

We compare the stability preserving properties of the Lie–Trotter, Strang–Marchuk, and symmetrically weighted sequential splitting schemes for a simple 2-dimensional linear system. We evaluate the trace and determinant of the split systems in terms the trace and determinant of the continuous system to establish stability criteria. We find that the stability region has a fingered structure, implying that stability is not a monotonic property of the splitting timestep. We provide estimates for the thickness of the stability fingers as well as the gaps between them. Counterintuitively, both the thickness and the size of gaps grow with decreasing splitting time step.