A hierarchical splitting approach for N-split differential equations

TL;DR

This paper introduces a hierarchical splitting framework for N-split differential equations, combining geometric and multirate integration techniques, with theoretical analysis and numerical validation demonstrating improved efficiency.

Contribution

It develops a novel hierarchical splitting method for N-split systems, analyzes its convergence and error properties, and integrates multirate techniques for enhanced computational performance.

Findings

Convergence order and error formulas derived for hierarchical splitting methods.

Conditions established for multirate factors to improve convergence.

Numerical experiments confirm theoretical predictions and efficiency gains.

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.