Operator splitting for exploiting linear-rate closure in solving infinite ODE hierarchies.

We introduce an operator-splitting method for infinite hierarchies of linear ordinary differential equations (ODEs) indexed by nonnegative integers. When the coupling coefficients depend linearly on the count index, an exact transformation closes the equations on finite count-index windows without an upper-boundary value. For more general hierarchies, Strang splitting applies the linear-rate closure during the linear-rate substeps and a conventional capped solver to the remainder. We derive the closure from generating functions and the method of characteristics and extend it to multi-indexed systems. The derivation requires neither positivity nor mass conservation, so it applies to a wider class of systems than the examplar stochastic models presented here. We discuss branching processes, stochastic predator-prey dynamics, the Schlögl chemical kinetics model, and a telegraph model for gene expression. Through numerical experiments and computational cost analyses we demonstrate that our operator splitting method is typically advantageous for solving large scale systems in terms of memory usage and computational time, while retaining accuracy competitive with finite state projection (FSP) methods.