A hierarchical splitting approach for N-split differential equations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917209901629440 |
|---|---|
| author | Schäfers, Kevin Günther, Michael |
| author_facet | Schäfers, Kevin Günther, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12878 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A hierarchical splitting approach for N-split differential equations Schäfers, Kevin Günther, Michael Numerical Analysis 65P10, 65L05, 65L20, 65Y20 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. |
| title | A hierarchical splitting approach for N-split differential equations |
| topic | Numerical Analysis 65P10, 65L05, 65L20, 65Y20 |
| url | https://arxiv.org/abs/2601.12878 |