Time integration algorithms with optimized controllable numerical dissipation for dynamic systems

Wang, Yazhou; Maxam, Dean; Liang, Xuan; Tamma, Kumar; Adams, Nikolaus

doi:10.1016/j.apm.2026.116878

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Title:: Time integration algorithms with optimized controllable numerical dissipation for dynamic systems
Document type:: Zeitschriftenaufsatz
Author(s):: Wang, Yazhou; Maxam, Dean; Liang, Xuan; Tamma, Kumar; Adams, Nikolaus
Abstract:: This paper presents a new family of implicit time integration algorithms with controllable numerical dissipation for first- and second-order initial value problems. The generalized time-weighted residual methodology provides the fundamental design framework, which is further investigated through cubic approximations to construct a new class of single-step algorithms with optimized numerical properties. Equivalent linear multistep formulations are also derived to enclose the design space. A non-standard stability analysis via the second-order equation of motion and a novel initialization strategy enable the proposed algorithms to achieve second-order accuracy in time, unconditional stability, zero-order overshooting, and controllable numerical dissipation and dispersion through a user-defined parameter, ρ∞, with optimal spectral accuracy. For the first time, the proposed computational framework unifies the L-stable Park method ((Formula presented) ) and the A-stable midpoint rule ((Formula presented) ), while introducing a spectrum of novel schemes for 0 < ρ∞ < 1, which do not require a distinct starting procedure as with their equivalent linear multistep counterparts. The enhanced dissipation control is validated through detailed comparisons with existing single-step and sub-step/multi-stage algorithms. Numerical examples involving both first- and second-order dynamic problems further demonstrate the improved numerical performance. The MATLAB code for the two-dimensional nonlinear heat transfer and three-dimensional solid mechanics problems is open-source to demonstrate the numerical implementations and compare the performance of various time integration algorithms in practical applications. © 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license. http://creativecommons.org/licenses/by/4.0/
Keywords:: Dynamic systems; Heat transfer; Optimized numerical dissipation; Structural dynamics; Time integration algorithms
Dewey Decimal Classification:: 620 Ingenieurwissenschaften
Journal title:: Applied Mathematical Modelling
Year:: 2026
Journal volume:: 156
Pages contribution:: 116878
Covered by:: Scopus
Language:: en
Fulltext / DOI:: doi:10.1016/j.apm.2026.116878
Publisher:: Elsevier BV
E-ISSN:: 0307-904X
Date of publication:: 01.08.2026
TUM Institution:: Lehrstuhl für Aerodynamik und Strömungsmechanik
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