A hierarchical splitting approach for N-split differential equations

TL;DR

The paper addresses designing high-order integrators for $N$-split differential equations by introducing hierarchical splitting, which recursively applies robust two-split solvers on a splitting tree. It shows that the order theory from two-split methods extends to arbitrary $N$-split systems, with order $p$ attained by assigning order-$p$ schemes to all nodes and supported by BCH/NB-series analyses and Gröbner’s Lemma, while preserving self-adjointness. A multirate extension with factors $M^{\{v\}}$ is developed, including reweighting to reduce computational overhead and a defined computational order that guarantees improved performance on practical step sizes $h\ge h_{\min}$. Numerical experiments on rigid body dynamics and a separable FPU system demonstrate the practical efficiency and accuracy of the hierarchical splitting framework, often outperforming naive composition approaches, and point to rich avenues for extending the approach to PDEs, DAEs, and SDEs as well as commutator-based variants. $N$-split problems in physics and engineering thus benefit from a principled, structure-preserving, multirate-capable tool for geometric numerical integration.

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.

Figures (9)

• Figure 1: A splitting tree for $N = 5$ partitions.
• Figure 1: Multirate splitting tree of a generalized impulse method for $N$-split systems.
• Figure 1: Rigid body. Global error at $t_{\mathtt{end}} = 100$ vs. step size $h$.
• Figure 2: Splitting tree for representing the Lie--Trotter splitting \ref{['eq:Lie-Trotter']} and the Strang splitting \ref{['eq:Strang-splitting']} as hierarchical splitting methods.
• Figure 2: Splitting trees for the rigid body equations.

Theorems & Definitions (7)

• Proof 1
• Proof 2
• Proof 3
• Proof 4
• Proof 5
• Proof 6
• Proof 7