A hierarchical splitting approach for N-split differential equations

Abstract: We propose a hierarchical splitting approach to differential equations that provides a design principle for constructing splitting methods for $N$-split systems by iteratively applying splitting methods for two-split systems. We analyze the convergence order, derive explicit formulas for the leading-order error terms, and investigate self-adjointness. Moreover, we discuss compositions of hierarchical splitting methods in detail. We further augment the hierarchical splitting approach with multiple time-stepping techniques, turning the class into a promising framework at the intersection of geometric numerical integration and multirate integration. In this context, we characterize the computational order of a multirate integrator and establish conditions on the multirate factors that guarantee an increased convergence rate in practical computations up to a certain step size. Finally, we design several hierarchical splitting methods and perform numerical simulations for rigid body equations and a separable Hamiltonian system with multirate potential, confirming the theoretical findings and showcasing the computational efficiency of hierarchical splitting methods.