Order elevation of directly self-starting sub-step implicit integrators for transient dynamics
Abstract
Directly self-starting implicit methods are attractive for transient analysis because they avoid auxiliary starting procedures while retaining the original first- or second-order governing equations. However, most existing formulations usually fix the last sub-step at the end of each time interval, which restricts the attainable order. This study develops a generalized s-sub-step implicit framework by releasing this constraint and treating all sub-step locations as design variables. The resulting methods admit a unified Runge--Kutta representation for both first- and second-order transient systems and preserve identical effective matrices over all sub-steps. Accuracy conditions are derived by simultaneously matching the numerical amplification factor and load operator, thereby accounting for both homogeneous and forced responses. For s=1, ⋯, 6, two complementary families are obtained: sth-order members with user-controllable high-frequency numerical dissipation and adjustable sub-step locations, and (s+1)th-order members obtained by selecting the sub-step locations, with fixed dissipation. The latter reach up to seventh-order accuracy without increasing the number of sub-steps, although some high-order members are A(α)-stable with stability angles extremely close to 90∘. Analytical amplitude and phase errors further reveal parity-dependent superconvergence in undamped systems, and appropriate parameter selections can substantially increase either phase or amplitude accuracy beyond the formal order. Numerical benchmarks confirm the predicted convergence orders and the controllable suppression of spurious high-frequency responses.
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BibTeX
@misc{li2026order,
title = {{Order elevation of directly self-starting sub-step implicit integrators for transient dynamics}},
author = {Jinze Li and Yaokun Liu and Kewei Chen and Hua Li and Kaiping Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10240},
eprint = {2609.10240},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10240},
}
RIS
TY - UNPB
TI - Order elevation of directly self-starting sub-step implicit integrators for transient dynamics
AU - Jinze Li
AU - Yaokun Liu
AU - Kewei Chen
AU - Hua Li
AU - Kaiping Yu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10240
UR - https://arxiv.org/abs/2609.10240
AB - Directly self-starting implicit methods are attractive for transient analysis because they avoid auxiliary starting procedures while retaining the original first- or second-order governing equations. However, most existing formulations usually fix the last sub-step at the end of each time interval, which restricts the attainable order. This study develops a generalized s-sub-step implicit framework by releasing this constraint and treating all sub-step locations as design variables. The resulting methods admit a unified Runge--Kutta representation for both first- and second-order transient systems and preserve identical effective matrices over all sub-steps. Accuracy conditions are derived by simultaneously matching the numerical amplification factor and load operator, thereby accounting for both homogeneous and forced responses. For s=1, ⋯, 6, two complementary families are obtained: sth-order members with user-controllable high-frequency numerical dissipation and adjustable sub-step locations, and (s+1)th-order members obtained by selecting the sub-step locations, with fixed dissipation. The latter reach up to seventh-order accuracy without increasing the number of sub-steps, although some high-order members are A(α)-stable with stability angles extremely close to 90^∘. Analytical amplitude and phase errors further reveal parity-dependent superconvergence in undamped systems, and appropriate parameter selections can substantially increase either phase or amplitude accuracy beyond the formal order. Numerical benchmarks confirm the predicted convergence orders and the controllable suppression of spurious high-frequency responses.
ER -